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Theorem en2d 6279
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 2056 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en2d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V )
)
54imp 119 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  _V )
6 en2d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V )
)
76imp 119 . . 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 5731 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
10 f1oen2g 6266 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
111, 2, 9, 10syl3anc 1146 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   class class class wbr 3792    |-> cmpt 3846   -1-1-onto->wf1o 4929    ~~ cen 6250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-en 6253
This theorem is referenced by:  en2i  6281
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