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Theorem f1oen3g 7528
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7531 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 5805 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
21spcegv 3199 . . 3  |-  ( F  e.  V  ->  ( F : A -1-1-onto-> B  ->  E. f 
f : A -1-1-onto-> B ) )
32imp 429 . 2  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  E. f 
f : A -1-1-onto-> B )
4 bren 7522 . 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 369   E.wex 1596    e. wcel 1767   class class class wbr 4447   -1-1-onto->wf1o 5585    ~~ cen 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-en 7514
This theorem is referenced by:  f1oen2g  7529  unen  7595  domdifsn  7597  domunsncan  7614  sbthlem10  7633  domssex  7675  phplem2  7694  sucdom2  7711  pssnn  7735  f1finf1o  7743  oien  7959  infdifsn  8069  fin4en1  8685  fin23lem21  8715  hashf1lem2  12465  odinf  16378  gsumval3OLD  16696  gsumval3lem1  16697  gsumval3lem2  16698  gsumval3  16699  hmphen2  20032
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