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Theorem foeq3 5313
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq3  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )

Proof of Theorem foeq3
StepHypRef Expression
1 eqeq2 2127 . . 3  |-  ( A  =  B  ->  ( ran  F  =  A  <->  ran  F  =  B ) )
21anbi2d 459 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  =  A )  <->  ( F  Fn  C  /\  ran  F  =  B ) ) )
3 df-fo 5099 . 2  |-  ( F : C -onto-> A  <->  ( F  Fn  C  /\  ran  F  =  A ) )
4 df-fo 5099 . 2  |-  ( F : C -onto-> B  <->  ( F  Fn  C  /\  ran  F  =  B ) )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   ran crn 4510    Fn wfn 5088   -onto->wfo 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-fo 5099
This theorem is referenced by:  f1oeq3  5328  foeq123d  5331  resdif  5357  ffoss  5367  fifo  6836  enumct  6968  ctssexmid  6992  exmidfodomrlemr  7026  exmidfodomrlemrALT  7027  qnnen  11871  unct  11881  subctctexmid  13123
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