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Theorem feq3 5380
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )

Proof of Theorem feq3
StepHypRef Expression
1 sseq2 3203 . . 3  |-  ( A  =  B  ->  ( ran  F  C_  A  <->  ran  F  C_  B ) )
21anbi2d 464 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  C_  A
)  <->  ( F  Fn  C  /\  ran  F  C_  B ) ) )
3 df-f 5250 . 2  |-  ( F : C --> A  <->  ( F  Fn  C  /\  ran  F  C_  A ) )
4 df-f 5250 . 2  |-  ( F : C --> B  <->  ( F  Fn  C  /\  ran  F  C_  B ) )
52, 3, 43bitr4g 223 1  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    C_ wss 3153   ran crn 4656    Fn wfn 5241   -->wf 5242
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166  df-f 5250
This theorem is referenced by:  feq23  5381  feq3d  5384  feq123d  5386  fun2  5419  fconstg  5442  f1eq3  5448  fsng  5723  fsn2  5724  fsnunf  5750  mapvalg  6703  mapsn  6735  lmff  14394  txcn  14420
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