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Theorem en3d 6898
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
en3d.1  |-  ( ph  ->  A  e.  _V )
en3d.2  |-  ( ph  ->  B  e.  _V )
en3d.3  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
en3d.4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
en3d.5  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
Assertion
Ref Expression
en3d  |-  ( 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 en3d
StepHypRef Expression
1 en3d.1 . 2  |-  ( ph  ->  A  e.  _V )
2 en3d.2 . 2  |-  ( ph  ->  B  e.  _V )
3 eqid 2283 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en3d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
54imp 418 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
6 en3d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
76imp 418 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
8 en3d.5 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
98imp 418 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
103, 5, 7, 9f1o2d 6069 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
11 f1oen2g 6878 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
121, 2, 10, 11syl3anc 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:  en3i  6900  fundmen  6934  mapen  7025  mapxpen  7027  mapunen  7030  ssenen  7035  fzen  10811  hashbclem  11390  hashfacen  11392  hashf1lem1  11393  hashdvds  12843  sylow2a  14930  lsmhash  15014  subfacp1lem3  23713  subfacp1lem5  23715
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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