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Theorem f1oen3g 6301
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6304 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 5148 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
21spcegv 2687 . . 3  |-  ( F  e.  V  ->  ( F : A -1-1-onto-> B  ->  E. f 
f : A -1-1-onto-> B ) )
32imp 122 . 2  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  E. f 
f : A -1-1-onto-> B )
4 bren 6294 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
53, 4sylibr 132 1  |-  ( ( F  e.  V  /\  F : A -1-1-onto-> B )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1422    e. wcel 1434   class class class wbr 3793   -1-1-onto->wf1o 4931    ~~ cen 6285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-en 6288
This theorem is referenced by:  f1oen2g  6302  unen  6361  phplem2  6388
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