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Theorem dom2 4386

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its range. and can be read and , as can be shown from their distinct variable conditions.

Hypotheses

Ref	Expression
dom2.1
dom2.2

Assertion

Ref	Expression
dom2

Distinct variable groups:   ,,   ,,   ,   ,

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Proof of Theorem dom2

Step	Hyp	Ref	Expression
1		eqid 1468	. 2
2		dom2.1	. . . 4
3	2	a1i 8	. . 3
4		dom2.2	. . . 4
5	4	a1i 8	. . 3
6	3, 5	dom2d 4385	. 2
7	1, 6	ax-mp 7	1

Colors of variables: wff set class

Syntax hints:   wi 3   wb 146   wa 223   wceq 953   wcel 955   class class class wbr 2609   cdom 4349

This theorem is referenced by:  canth2 4464  limenpsi 4485  xpnnen 7441  znnen 7445

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-9 962  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-rep 2683  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-ral 1641  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-rel 3175  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-en 4351  df-dom 4352

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