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Theorem en3d 6941
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 2316 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en3d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
54imp 418 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
6 en3d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
76imp 418 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
8 en3d.5 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
98imp 418 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
103, 5, 7, 9f1o2d 6111 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
11 f1oen2g 6921 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
121, 2, 10, 11syl3anc 1182 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   class class class wbr 4060    e. cmpt 4114   -1-1-onto->wf1o 5291    ~~ cen 6903
This theorem is referenced by:  en3i  6943  fundmen  6977  mapen  7068  mapxpen  7070  mapunen  7073  ssenen  7078  fzen  10858  hashbclem  11437  hashfacen  11439  hashf1lem1  11440  hashdvds  12890  sylow2a  14979  lsmhash  15063  subfacp1lem3  23997  subfacp1lem5  23999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-en 6907
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