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Theorem dom2d 7576
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2d.1
dom2d.2
Assertion
Ref Expression
dom2d
Distinct variable groups:   , ,   , ,   ,   ,   , ,

Proof of Theorem dom2d
StepHypRef Expression
1 dom2d.1 . . 3
2 dom2d.2 . . 3
31, 2dom2lem 7575 . 2
4 f1domg 7555 . 2
53, 4syl5com 30 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   class class class wbr 4452  e.cmpt 4510  -1-1->wf1 5590   cdom 7534
This theorem is referenced by:  dom2  7578  fineqvlem  7754  fseqdom  8428  fin1a2lem9  8809  iundom2g  8936  canthwe  9050  prmreclem2  14435  prmreclem3  14436  sylow1lem4  16621  aannenlem1  22724  derangenlem  28615  fphpd  30750  pellexlem3  30767  unxpwdom3  31041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-dom 7538
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