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Theorem feq3 5332
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 3171 . . 3  |-  ( A  =  B  ->  ( ran  F  C_  A  <->  ran  F  C_  B ) )
21anbi2d 461 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  C_  A
)  <->  ( F  Fn  C  /\  ran  F  C_  B ) ) )
3 df-f 5202 . 2  |-  ( F : C --> A  <->  ( F  Fn  C  /\  ran  F  C_  A ) )
4 df-f 5202 . 2  |-  ( F : C --> B  <->  ( F  Fn  C  /\  ran  F  C_  B ) )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    C_ wss 3121   ran crn 4612    Fn wfn 5193   -->wf 5194
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202
This theorem is referenced by:  feq23  5333  feq3d  5336  feq123d  5338  fun2  5371  fconstg  5394  f1eq3  5400  fsng  5669  fsn2  5670  fsnunf  5696  mapvalg  6636  mapsn  6668  lmff  13043  txcn  13069
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