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Theorem dom2d 6902
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 6901 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
4 f1domg 6881 . 2  |-  ( B  e.  R  ->  (
( x  e.  A  |->  C ) : A -1-1-> B  ->  A  ~<_  B ) )
53, 4syl5com 26 1  |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   -1-1->wf1 5252    ~<_ cdom 6861
This theorem is referenced by:  dom2  6904  fineqvlem  7077  fseqdom  7653  fin1a2lem9  8034  iundom2g  8162  canthwe  8273  prmreclem2  12964  prmreclem3  12965  sylow1lem4  14912  aannenlem1  19708  derangenlem  23702  fphpd  26899  pellexlem3  26916  unxpwdom3  27256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dom 6865
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