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Theorem en2d 7146
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 2438 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en2d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V )
)
54imp 420 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  _V )
6 en2d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V )
)
76imp 420 . . 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 6297 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
10 f1oen2g 7127 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
111, 2, 9, 10syl3anc 1185 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   class class class wbr 4215    e. cmpt 4269   -1-1-onto->wf1o 5456    ~~ cen 7109
This theorem is referenced by:  en2i  7148  map1  7188  gicsubgen  15070  lzenom  26842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-en 7113
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