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Theorem feq3 5304
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 3152 . . 3  |-  ( A  =  B  ->  ( ran  F  C_  A  <->  ran  F  C_  B ) )
21anbi2d 460 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  C_  A
)  <->  ( F  Fn  C  /\  ran  F  C_  B ) ) )
3 df-f 5174 . 2  |-  ( F : C --> A  <->  ( F  Fn  C  /\  ran  F  C_  A ) )
4 df-f 5174 . 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 1335    C_ wss 3102   ran crn 4587    Fn wfn 5165   -->wf 5166
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-f 5174
This theorem is referenced by:  feq23  5305  feq3d  5308  feq123d  5310  fun2  5343  fconstg  5366  f1eq3  5372  fsng  5640  fsn2  5641  fsnunf  5667  mapvalg  6603  mapsn  6635  lmff  12660  txcn  12686
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