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Theorem en2d 6767
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 2177 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en2d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V )
)
54imp 124 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  _V )
6 en2d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V )
)
76imp 124 . . 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 6073 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
10 f1oen2g 6754 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
111, 2, 9, 10syl3anc 1238 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737   class class class wbr 4003    |-> cmpt 4064   -1-1-onto->wf1o 5215    ~~ cen 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-en 6740
This theorem is referenced by:  en2i  6769  map1  6811
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