MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oen3g Structured version   Unicode version

Theorem f1oen3g 7569
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7572 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen3g  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )

Proof of Theorem f1oen3g
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 f1oeq1 5790 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
21spcegv 3145 . . 3  |-  ( F  e.  V  ->  ( F : A -1-1-onto-> B  ->  E. f 
f : A -1-1-onto-> B ) )
32imp 427 . 2  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  E. f 
f : A -1-1-onto-> B )
4 bren 7563 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
53, 4sylibr 212 1  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1633    e. wcel 1842   class class class wbr 4395   -1-1-onto->wf1o 5568    ~~ cen 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-en 7555
This theorem is referenced by:  f1oen2g  7570  unen  7636  domdifsn  7638  domunsncan  7655  sbthlem10  7674  domssex  7716  phplem2  7735  sucdom2  7751  pssnn  7773  f1finf1o  7781  oien  7997  infdifsn  8106  fin4en1  8721  fin23lem21  8751  hashf1lem2  12554  odinf  16909  gsumval3OLD  17232  gsumval3lem1  17233  gsumval3lem2  17234  gsumval3  17235  hmphen2  20592
  Copyright terms: Public domain W3C validator