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Theorem en3d 7606
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 2451 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en3d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
54imp 431 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
6 en3d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
76imp 431 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
8 en3d.5 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
98imp 431 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
103, 5, 7, 9f1o2d 6521 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
11 f1oen2g 7586 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
121, 2, 10, 11syl3anc 1268 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   class class class wbr 4402    |-> cmpt 4461   -1-1-onto->wf1o 5581    ~~ cen 7566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-en 7570
This theorem is referenced by:  en3i  7608  fundmen  7643  mapen  7736  mapxpen  7738  mapunen  7741  ssenen  7746  fzen  11816  hashbclem  12615  hashfacen  12617  hashf1lem1  12618  hashdvds  14723  sylow2a  17271  lsmhash  17355  subfacp1lem3  29905  subfacp1lem5  29907
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