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Theorem f1oeq3 5119
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 5089 . . 3  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
2 foeq3 5104 . . 3  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
31, 2anbi12d 442 . 2  |-  ( A  =  B  ->  (
( F : C -1-1-> A  /\  F : C -onto-> A )  <->  ( F : C -1-1-> B  /\  F : C -onto-> B ) ) )
4 df-f1o 4909 . 2  |-  ( F : C -1-1-onto-> A  <->  ( F : C -1-1-> A  /\  F : C -onto-> A ) )
5 df-f1o 4909 . 2  |-  ( F : C -1-1-onto-> B  <->  ( F : C -1-1-> B  /\  F : C -onto-> B ) )
63, 4, 53bitr4g 212 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 97    <-> wb 98    = wceq 1243   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  f1oeq23  5120  f1oeq123d  5123  f1ores  5141  resdif  5148  f1osng  5167  f1oresrab  5329  isoeq5  5445  isoini2  5458  bren  6228  xpcomf1o  6299  frechashgf1o  9179  sumeq1  9848
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