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Theorem fneq1 4933
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 4867 . . 3  F  G  Fun  F  Fun  G
2 dmeq 4481 . . . 4  F  G  dom  F  dom  G
32eqeq1d 2048 . . 3  F  G  dom  F  dom  G
41, 3anbi12d 442 . 2  F  G  Fun  F  dom  F  Fun 
G  dom  G
5 df-fn 4851 . 2  F  Fn  Fun 
F  dom  F
6 df-fn 4851 . 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 1243   dom cdm 4291   Fun wfun 4842    Fn wfn 4843
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-fun 4850  df-fn 4851
This theorem is referenced by:  fneq1d  4935  fneq1i  4939  fn0  4964  feq1  4976  foeq1  5048  f1ocnv  5085  mpteqb  5207  eufnfv  5335  tfr0  5882  tfrlemiex  5890
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