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Theorem en3d 7589
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 2402 . . 3  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
4 en3d.3 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )
54imp 427 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  B )
6 en3d.4 . . . 4  |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )
76imp 427 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  D  e.  A )
8 en3d.5 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  =  D  <->  y  =  C ) ) )
98imp 427 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  -> 
( x  =  D  <-> 
y  =  C ) )
103, 5, 7, 9f1o2d 6507 . 2  |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-onto-> B
)
11 f1oen2g 7569 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  (
x  e.  A  |->  C ) : A -1-1-onto-> B )  ->  A  ~~  B
)
121, 2, 10, 11syl3anc 1230 1  |-  ( ph  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   class class class wbr 4394    |-> cmpt 4452   -1-1-onto->wf1o 5567    ~~ cen 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-en 7554
This theorem is referenced by:  en3i  7591  fundmen  7626  mapen  7718  mapxpen  7720  mapunen  7723  ssenen  7728  fzen  11755  hashbclem  12548  hashfacen  12550  hashf1lem1  12551  hashdvds  14512  sylow2a  16961  lsmhash  17045  subfacp1lem3  29466  subfacp1lem5  29468
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