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Theorem fneq1 4930
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
fneq1  F  G  F  Fn  G  Fn

Proof of Theorem fneq1
StepHypRef Expression
1 funeq 4864 . . 3  F  G  Fun  F  Fun  G
2 dmeq 4478 . . . 4  F  G  dom  F  dom  G
32eqeq1d 2045 . . 3  F  G  dom  F  dom  G
41, 3anbi12d 442 . 2  F  G  Fun  F  dom  F  Fun 
G  dom  G
5 df-fn 4848 . 2  F  Fn  Fun 
F  dom  F
6 df-fn 4848 . 2  G  Fn  Fun 
G  dom  G
74, 5, 63bitr4g 212 1  F  G  F  Fn  G  Fn
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   dom cdm 4288   Fun wfun 4839    Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  fneq1d  4932  fneq1i  4936  fn0  4961  feq1  4973  foeq1  5045  f1ocnv  5082  mpteqb  5204  eufnfv  5332  tfr0  5878  tfrlemiex  5886
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