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Theorem fvelrn 3797
Description: A function's value belongs to its range.
Assertion
Ref Expression
fvelrn |- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Proof of Theorem fvelrn
StepHypRef Expression
1 eleq1 1526 . . . . 5 |- (x = A -> (x e. dom F <-> A e. dom F))
21anbi2d 614 . . . 4 |- (x = A -> ((Fun F /\ x e. dom F) <-> (Fun F /\ A e. dom F)))
3 fveq2 3709 . . . . 5 |- (x = A -> (F` x) = (F` A))
43eleq1d 1532 . . . 4 |- (x = A -> ((F` x) e. ran F <-> (F` A) e. ran F))
52, 4imbi12d 624 . . 3 |- (x = A -> (((Fun F /\ x e. dom F) -> (F` x) e. ran F) <-> ((Fun F /\ A e. dom F) -> (F` A) e. ran F)))
6 funfvop 3788 . . . . 5 |- ((Fun F /\ x e. dom F) -> <.x, (F` x)>. e. F)
7 visset 1804 . . . . . 6 |- x e. V
8 opeq1 2478 . . . . . . 7 |- (y = x -> <.y, (F` x)>. = <.x, (F` x)>.)
98eleq1d 1532 . . . . . 6 |- (y = x -> (<.y, (F` x)>. e. F <-> <.x, (F` x)>. e. F))
107, 9cla4ev 1860 . . . . 5 |- (<.x, (F` x)>. e. F -> E.y<.y, (F` x)>. e. F)
116, 10syl 10 . . . 4 |- ((Fun F /\ x e. dom F) -> E.y<.y, (F` x)>. e. F)
12 fvex 3717 . . . . 5 |- (F` x) e. V
1312elrn2 3335 . . . 4 |- ((F` x) e. ran F <-> E.y<.y, (F` x)>. e. F)
1411, 13sylibr 200 . . 3 |- ((Fun F /\ x e. dom F) -> (F` x) e. ran F)
155, 14vtoclg 1838 . 2 |- (A e. dom F -> ((Fun F /\ A e. dom F) -> (F` A) e. ran F))
1615anabsi7 496 1 |- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  <.cop 2401  dom cdm 3160  ran crn 3161  Fun wfun 3166  ` cfv 3172
This theorem is referenced by:  fnfvelrn 3798  funfvima 3837  elunirnALT 3854  tz7.48-2 3942  fnoprvalrn2 10366  rdmob 10525  rcmob 10526
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188
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