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Theorem f1oen2g 6894
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6896 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 5488 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
2 fex2 5417 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1215 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
433coml 1158 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F  e.  _V )
5 simp3 957 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F : A -1-1-onto-> B )
6 f1oen3g 6893 . 2  |-  ( ( F  e.  _V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
74, 5, 6syl2anc 642 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   -->wf 5267   -1-1-onto->wf1o 5270    ~~ cen 6876
This theorem is referenced by:  f1oeng  6896  enrefg  6909  en2d  6913  en3d  6914  ener  6924  f1imaen2g  6938  cnven  6952  xpcomen  6969  omxpen  6980  pw2eng  6984  unfilem3  7139  xpfi  7144  hsmexlem1  8068  iccen  10795  uzenom  11043  nnenom  11058  eqgen  14686  dfod2  14893  hmphen  17492  0sgmppw  20453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6880
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