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Theorem f1oen2g 7488
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7490 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 5753 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
2 fex2 6691 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1261 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
433coml 1202 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F  e.  _V )
5 simp3 997 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F : A -1-1-onto-> B )
6 f1oen3g 7487 . 2  |-  ( ( F  e.  _V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
74, 5, 6syl2anc 659 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 972    e. wcel 1840   _Vcvv 3056   class class class wbr 4392   -->wf 5519   -1-1-onto->wf1o 5522    ~~ cen 7469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-en 7473
This theorem is referenced by:  f1oeng  7490  enrefg  7503  en2d  7507  en3d  7508  ener  7518  f1imaen2g  7532  cnven  7547  xpcomen  7564  omxpen  7575  pw2eng  7579  unfilem3  7738  xpfi  7743  hsmexlem1  8756  iccen  11634  uzenom  12027  nnenom  12042  eqgen  16468  dfod2  16800  hmphen  20468  0sgmppw  23744
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