MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oen2g Structured version   Unicode version

Theorem f1oen2g 7533
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7535 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 5767 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
2 fex2 6699 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1297 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
433coml 1212 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F  e.  _V )
5 simp3 1007 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-onto-> B )  ->  F : A -1-1-onto-> B )
6 f1oen3g 7532 . 2  |-  ( ( F  e.  _V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
74, 5, 6syl2anc 665 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 982    e. wcel 1872   _Vcvv 3016   class class class wbr 4359   -->wf 5533   -1-1-onto->wf1o 5536    ~~ cen 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-en 7518
This theorem is referenced by:  f1oeng  7535  enrefg  7548  en2d  7552  en3d  7553  ener  7563  f1imaen2g  7577  cnven  7592  xpcomen  7609  omxpen  7620  pw2eng  7624  unfilem3  7783  xpfi  7788  hsmexlem1  8800  iccen  11721  uzenom  12121  nnenom  12136  eqgen  16806  dfod2  17151  hmphen  20735  0sgmppw  24061
  Copyright terms: Public domain W3C validator