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Theorem en2d 6897
Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
Hypotheses
Ref Expression
en2d.1  |-  ( ph  ->  A  e.  _V )
en2d.2  |-  ( ph  ->  B  e.  _V )
en2d.3  |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V )
)
en2d.4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V )
)
en2d.5  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
Assertion
Ref Expression
en2d  |-  ( ph  ->  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)

Proof of Theorem en2d
StepHypRef Expression
1 en2d.1 . 2  |-  ( ph  ->  A  e.  _V )
2 en2d.2 . 2  |-  ( ph  ->  B  e.  _V )
3 eqid 2283 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en2d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V )
)
54imp 418 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  _V )
6 en2d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V )
)
76imp 418 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  _V )
8 en2d.5 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) ) )
93, 5, 7, 8f1od 6067 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
10 f1oen2g 6878 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
111, 2, 9, 10syl3anc 1182 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   -1-1-onto->wf1o 5254    ~~ cen 6860
This theorem is referenced by:  en2i  6899  map1  6939  gicsubgen  14742  lzenom  26849
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-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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864
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