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Theorem feq3 5377
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 3200 . . 3  |-  ( A  =  B  ->  ( ran  F  C_  A  <->  ran  F  C_  B ) )
21anbi2d 684 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  C_  A
)  <->  ( F  Fn  C  /\  ran  F  C_  B ) ) )
3 df-f 5259 . 2  |-  ( F : C --> A  <->  ( F  Fn  C  /\  ran  F  C_  A ) )
4 df-f 5259 . 2  |-  ( F : C --> B  <->  ( F  Fn  C  /\  ran  F  C_  B ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152   ran crn 4690    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  feq23  5378  feq123d  5382  fun2  5406  fconstg  5428  f1eq3  5434  fsng  5697  fsn2  5698  fsnunf  5718  mapvalg  6782  mapsn  6809  cantnff  7375  axdc4uz  11045  supcvg  12314  funcres2b  13771  funcres2  13772  hofcl  14033  resmhm2b  14438  pwsdiagmhm  14445  gsumress  14454  frmdup3  14488  isghm  14683  frgpup3lem  15086  gsumzsubmcl  15200  dmdprd  15236  cnpf2  16980  lmff  17029  2ndcctbss  17181  1stcelcls  17187  uptx  17319  txcn  17320  tsmssubm  17825  pi1addf  18545  lmmbr  18684  caufval  18701  iscmet3  18719  equivcau  18726  lmcau  18738  dvcnvrelem2  19365  itgsubstlem  19395  plypf1  19594  coef2  19613  ulmval  19759  isgrpo  20863  elghomlem1  21028  vci  21104  isvclem  21133  vcoprnelem  21134  chscllem4  22219  nmop0h  22571  cvmliftlem15  23829  isumgra  23867  iseupa  23881  ghomgrpilem2  23993  limptlimpr2lem1  25574  limptlimpr2lem2  25575  flfneic  25624  ismgra  25710  isalg  25721  algi  25727  aidm2  25750  dualalg  25782  sdclem1  26453  isbnd3  26508  prdsbnd  26517  heibor  26545  frlmup2  27251  stoweidlem57  27806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-f 5259
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