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Theorem f1oen2g 7124
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7126 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 5674 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
2 fex2 5603 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1217 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
433coml 1160 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F  e.  _V )
5 simp3 959 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F : A -1-1-onto-> B )
6 f1oen3g 7123 . 2  |-  ( ( F  e.  _V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
74, 5, 6syl2anc 643 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 936    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   -->wf 5450   -1-1-onto->wf1o 5453    ~~ cen 7106
This theorem is referenced by:  f1oeng  7126  enrefg  7139  en2d  7143  en3d  7144  ener  7154  f1imaen2g  7168  cnven  7182  xpcomen  7199  omxpen  7210  pw2eng  7214  unfilem3  7373  xpfi  7378  hsmexlem1  8306  iccen  11040  uzenom  11304  nnenom  11319  eqgen  14993  dfod2  15200  hmphen  17817  0sgmppw  20982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-en 7110
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