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Theorem en3d 6796
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 2189 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en3d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
54imp 124 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
6 en3d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
76imp 124 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
8 en3d.5 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
98imp 124 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
103, 5, 7, 9f1o2d 6100 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
11 f1oen2g 6782 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
121, 2, 10, 11syl3anc 1249 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   class class class wbr 4018    |-> cmpt 4079   -1-1-onto->wf1o 5234    ~~ cen 6765
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-en 6768
This theorem is referenced by:  en3i  6798  fundmen  6833  mapen  6875  mapxpen  6877  ssenen  6880  fzen  10075  uzennn  10469  hashfacen  10851  hashdvds  12256
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