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Theorem feq3 5225
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 3089 . . 3  |-  ( A  =  B  ->  ( ran  F  C_  A  <->  ran  F  C_  B ) )
21anbi2d 457 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  C_  A
)  <->  ( F  Fn  C  /\  ran  F  C_  B ) ) )
3 df-f 5095 . 2  |-  ( F : C --> A  <->  ( F  Fn  C  /\  ran  F  C_  A ) )
4 df-f 5095 . 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 1314    C_ wss 3039   ran crn 4508    Fn wfn 5086   -->wf 5087
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052  df-f 5095
This theorem is referenced by:  feq23  5226  feq3d  5229  feq123d  5231  fun2  5264  fconstg  5287  f1eq3  5293  fsng  5559  fsn2  5560  fsnunf  5586  mapvalg  6518  mapsn  6550  lmff  12313  txcn  12339
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