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Theorem dom2d 7342
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  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
dom2d.2  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
Assertion
Ref Expression
dom2d  |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D    ph, x, y
Allowed substitution hints:    C( x)    D( y)    R( x, y)

Proof of Theorem dom2d
StepHypRef Expression
1 dom2d.1 . . 3  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
2 dom2d.2 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )
31, 2dom2lem 7341 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1domg 7321 . 2  |-  ( B  e.  R  ->  (
( x  e.  A  |->  C ) : A -1-1-> B  ->  A  ~<_  B ) )
53, 4syl5com 30 1  |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4287    e. cmpt 4345   -1-1->wf1 5410    ~<_ cdom 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-dom 7304
This theorem is referenced by:  dom2  7344  fineqvlem  7519  fseqdom  8188  fin1a2lem9  8569  iundom2g  8696  canthwe  8810  prmreclem2  13970  prmreclem3  13971  sylow1lem4  16091  aannenlem1  21769  derangenlem  27011  fphpd  29108  pellexlem3  29125  unxpwdom3  29401
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