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Theorem en2d 7040
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 2366 . . 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 6194 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
10 f1oen2g 7021 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
111, 2, 9, 10syl3anc 1183 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   class class class wbr 4125    e. cmpt 4179   -1-1-onto->wf1o 5357    ~~ cen 7003
This theorem is referenced by:  en2i  7042  map1  7082  gicsubgen  14952  lzenom  26440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-en 7007
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