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Theorem f1oeq3 5453
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq3  |-  ( A  =  B  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )

Proof of Theorem f1oeq3
StepHypRef Expression
1 f1eq3 5420 . . 3  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
2 foeq3 5438 . . 3  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
31, 2anbi12d 473 . 2  |-  ( A  =  B  ->  (
( F : C -1-1-> A  /\  F : C -onto-> A )  <->  ( F : C -1-1-> B  /\  F : C -onto-> B ) ) )
4 df-f1o 5225 . 2  |-  ( F : C -1-1-onto-> A  <->  ( F : C -1-1-> A  /\  F : C -onto-> A ) )
5 df-f1o 5225 . 2  |-  ( F : C -1-1-onto-> B  <->  ( F : C -1-1-> B  /\  F : C -onto-> B ) )
63, 4, 53bitr4g 223 1  |-  ( A  =  B  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   -1-1->wf1 5215   -onto->wfo 5216   -1-1-onto->wf1o 5217
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225
This theorem is referenced by:  f1oeq23  5454  f1oeq123d  5457  f1oeq3d  5460  f1ores  5478  resdif  5485  f1osng  5504  f1oresrab  5683  isoeq5  5808  isoini2  5822  mapsnf1o  6739  bren  6749  xpcomf1o  6827  frechashgf1o  10430  sumeq1  11365  fisumss  11402  fsumcnv  11447  prodeq1f  11562  ennnfonelemhf1o  12416  ennnfonelemex  12417  ssnnctlemct  12449
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