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Theorem feq1 5081
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
feq1  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )

Proof of Theorem feq1
StepHypRef Expression
1 fneq1 5038 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 4609 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32sseq1d 3035 . . 3  |-  ( F  =  G  ->  ( ran  F  C_  B  <->  ran  G  C_  B ) )
41, 3anbi12d 457 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  C_  B
)  <->  ( G  Fn  A  /\  ran  G  C_  B ) ) )
5 df-f 4956 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
6 df-f 4956 . 2  |-  ( G : A --> B  <->  ( G  Fn  A  /\  ran  G  C_  B ) )
74, 5, 63bitr4g 221 1  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    C_ wss 2982   ran crn 4392    Fn wfn 4947   -->wf 4948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  feq1d  5085  feq1i  5090  f00  5132  fconstg  5134  f1eq1  5138  fconst2g  5428  tfrcllemsucfn  6022  tfrcllemsucaccv  6023  tfrcllembxssdm  6025  tfrcllembfn  6026  tfrcllemex  6029  tfrcllemaccex  6030  tfrcllemres  6031  tfrcl  6033  ac6sfi  6454  1fv  9278
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