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Statement List for Metamath Proof Explorer - 3401-3500 - Page 35 of 107

Type	Label	Description
Statement
Theorem	dfima2 3401	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
Theorem	dfima3 3402	Alternate definition of image. Compare definition (d) of [Enderton] p. 44.
Theorem	elimag 3403	Membership in an image. Theorem 34 of [Suppes] p. 65.
Theorem	elima 3404	Membership in an image. Theorem 34 of [Suppes] p. 65.
Theorem	elima2 3405	Membership in an image. Theorem 34 of [Suppes] p. 65.
Theorem	elima3 3406	Membership in an image. Theorem 34 of [Suppes] p. 65.
Theorem	hbima 3407	Bound-variable hypothesis builder for image.
Theorem	hbimad 3408	Deduction version of bound-variable hypothesis builder hbima 3407. (Contributed by FL, 15-Dec-2006.)
Theorem	csbima12g 3409	Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.)
Theorem	imadmrn 3410	The image of the domain of a class is the range of the class.
Theorem	imassrn 3411	The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39.
Theorem	imaexg 3412	The image of a set is a set. Theorem 3.17 of [Monk1] p. 39.
Theorem	imai 3413	Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38.
Theorem	rnresi 3414	The range of the restricted identity function.
Theorem	resiima 3415	The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
Theorem	ima0 3416	Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38.
Theorem	0ima 3417	Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
Theorem	imadisj 3418	A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
Theorem	cnvimass 3419	A pre-image under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
Theorem	imasng 3420	The image of a singleton.
Theorem	relimasn 3421	The image of a singleton.
Theorem	elimasn 3422	Membership in an image of a singleton.
Theorem	elimasng 3423	Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
Theorem	args 3424	Two ways to express the class of unique-valued arguments of , which is the same as the domain of whenever is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg " for this class (for which we have no separate notation). Observe the resemblance to our df-fv 3197, which was based on the idea in Quine's definition.
Theorem	eliniseg 3425	Membership in an initial segment. The idiom , meaning , is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30.
Theorem	iniseg 3426	An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30.
Theorem	dffr3 3427	Alternate definition of founded relation. Definition 6.21 of [TakeutiZaring] p. 30.
Theorem	imass1 3428	Subset theorem for image.
Theorem	imass2 3429	Subset theorem for image. Exercise 22(a) of [Enderton] p. 53.
Theorem	ndmima 3430	The image of a singleton outside the domain is empty.
Theorem	relcnv 3431	A converse is a relation. Theorem 12 of [Suppes] p. 62.
Theorem	cotr 3432	Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51.
Theorem	cnvsym 3433	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51.
Theorem	intasym 3434	Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51.
Theorem	asymref 3435	Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 3514.
Theorem	asymref2 3436	Two ways of saying a relation is antisymmetric and reflexive.
Theorem	asymrefOLD 3437	Two ways of saying a relation is antisymmetric and reflexive.
Theorem	asymref2OLD 3438	Two ways of saying a relation is antisymmetric and reflexive.
Theorem	intirr 3439	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51.
Theorem	soirri 3440	A strict order relation is irreflexive.
Theorem	sotri 3441	A strict order relation is a transitive relation.
Theorem	son2lpi 3442	A strict order relation has no 2-cycle loops.
Theorem	cnvopab 3443	The converse of a class abstraction of ordered pairs.
Theorem	cnv0 3444	The converse of the empty set.
Theorem	cnvi 3445	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36.
Theorem	op1sta 3446	Extract the first member of an ordered pair. (See op2nda 3450 to extract the second member, op1stb 2911 for an alternate version, and op1st 4084 for the preferred version..) (Contributed by Raph Levien, 4-Dec-2003.)
Theorem	cnvsn 3447	Converse of a singleton of an ordered pair.
Theorem	rnsnop 3448	The range of a singleton of an ordered pair is the singleton of the second member.
Theorem	op2ndb 3449	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 2911 to extract the first member, op2nda 3450 for an alternate version, and op2nd 4085 for the preferred version.)
Theorem	op2nda 3450	Extract the second member of an ordered pair. (See op1sta 3446 to extract the first member, op2ndb 3449 for an alternate version, and op2nd 4085 for the preferred version.)
Theorem	elxp4 3451	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 3452, elxp6 4101, and elxp7 4102.
Theorem	elxp5 3452	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 3451 when the double intersection does not create class existence problems (caused by int0 2543).
Theorem	cnvun 3453	The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62.
Theorem	cnvin 3454	Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62.
Theorem	rnun 3455	Distributive law for range over union. Theorem 8 of [Suppes] p. 60.
Theorem	rnin 3456	The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60.
Theorem	rnuni 3457	The range of a union. Part of Exercise 8 of [Enderton] p. 41.
Theorem	imaun 3458	Distributive law for image over union. Theorem 35 of [Suppes] p. 65.
Theorem	imaun2 3459	The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
Theorem	dminss 3460	An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising."
Theorem	imainss 3461	An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66.
Theorem	cnvxp 3462