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Theorem f1oen2g 7439
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7441 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen2g  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )

Proof of Theorem f1oen2g
StepHypRef Expression
1 f1of 5752 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
2 fex2 6645 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1252 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
433coml 1195 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F  e.  _V )
5 simp3 990 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F : A -1-1-onto-> B )
6 f1oen3g 7438 . 2  |-  ( ( F  e.  _V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
74, 5, 6syl2anc 661 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    e. wcel 1758   _Vcvv 3078   class class class wbr 4403   -->wf 5525   -1-1-onto->wf1o 5528    ~~ cen 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-en 7424
This theorem is referenced by:  f1oeng  7441  enrefg  7454  en2d  7458  en3d  7459  ener  7469  f1imaen2g  7483  cnven  7498  xpcomen  7515  omxpen  7526  pw2eng  7530  unfilem3  7692  xpfi  7697  hsmexlem1  8709  iccen  11550  uzenom  11907  nnenom  11922  eqgen  15856  dfod2  16189  hmphen  19493  0sgmppw  22673
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