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Theorem foeq3 5449
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 2292 . . 3  |-  ( A  =  B  ->  ( ran  F  =  A  <->  ran  F  =  B ) )
21anbi2d 684 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  =  A )  <->  ( F  Fn  C  /\  ran  F  =  B ) ) )
3 df-fo 5261 . 2  |-  ( F : C -onto-> A  <->  ( F  Fn  C  /\  ran  F  =  A ) )
4 df-fo 5261 . 2  |-  ( F : C -onto-> B  <->  ( F  Fn  C  /\  ran  F  =  B ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   ran crn 4690    Fn wfn 5250   -onto->wfo 5253
This theorem is referenced by:  f1oeq3  5465  foeq123d  5468  resdif  5494  ffoss  5505  fidomdm  7138  fifo  7185  brwdom  7281  brwdom2  7287  canthwdom  7293  ixpiunwdom  7305  fin1a2lem7  8032  znnen  12491  divslem  13445  znzrhfo  16501  rncmp  17123  conima  17151  concn  17152  qtopcmplem  17398  qtoprest  17408  opidon2  20991  pjhfo  22285  dmct  23342  ivthALT  26258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2276  df-fo 5261
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