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Theorem enssdom 7134
 Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
enssdom

Proof of Theorem enssdom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relen 7116 . 2
2 f1of1 5675 . . . . 5
32eximi 1586 . . . 4
4 opabid 4463 . . . 4
5 opabid 4463 . . . 4
63, 4, 53imtr4i 259 . . 3
7 df-en 7112 . . . 4
87eleq2i 2502 . . 3
9 df-dom 7113 . . . 4
109eleq2i 2502 . . 3
116, 8, 103imtr4i 259 . 2
121, 11relssi 4969 1
 Colors of variables: wff set class Syntax hints:  wex 1551   wcel 1726   wss 3322  cop 3819  copab 4267  wf1 5453  wf1o 5455   cen 7108   cdom 7109 This theorem is referenced by:  dfdom2  7135  endom  7136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887  df-f1o 5463  df-en 7112  df-dom 7113
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