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Theorem f1oen3g 6747
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6750 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 5444 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
21spcegv 2825 . . 3  |-  ( F  e.  V  ->  ( F : A -1-1-onto-> B  ->  E. f 
f : A -1-1-onto-> B ) )
32imp 124 . 2  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  E. f 
f : A -1-1-onto-> B )
4 bren 6740 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
53, 4sylibr 134 1  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1492    e. wcel 2148   class class class wbr 4000   -1-1-onto->wf1o 5210    ~~ cen 6731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-en 6734
This theorem is referenced by:  f1oen2g  6748  unen  6809  phplem2  6846  sbthlemi10  6958
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