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Theorem feq1 5223
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 5179 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 4734 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32sseq1d 3094 . . 3  |-  ( F  =  G  ->  ( ran  F  C_  B  <->  ran  G  C_  B ) )
41, 3anbi12d 462 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  C_  B
)  <->  ( G  Fn  A  /\  ran  G  C_  B ) ) )
5 df-f 5095 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
6 df-f 5095 . 2  |-  ( G : A --> B  <->  ( G  Fn  A  /\  ran  G  C_  B ) )
74, 5, 63bitr4g 222 1  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> 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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  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-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095
This theorem is referenced by:  feq1d  5227  feq1i  5233  f00  5282  f0bi  5283  f0dom0  5284  fconstg  5287  f1eq1  5291  fconst2g  5601  tfrcllemsucfn  6216  tfrcllemsucaccv  6217  tfrcllembxssdm  6219  tfrcllembfn  6220  tfrcllemex  6223  tfrcllemaccex  6224  tfrcllemres  6225  tfrcl  6227  elmapg  6521  ac6sfi  6758  updjud  6933  finomni  6978  exmidomni  6980  mkvprop  6998  1fv  9867  upxp  12347  txcn  12350
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