Theorem fneq1 5038

Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
fneq1	|- ( F = G -> ( F Fn A <-> G Fn A ) )

Proof of Theorem fneq1

Step	Hyp	Ref	Expression
1		funeq 4971	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4583	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2091	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 457	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 4955	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 4955	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 221	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   dom cdm 4391   Fun wfun 4946   Fn wfn 4947

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-fun 4954  df-fn 4955

This theorem is referenced by:  fneq1d  5040  fneq1i  5044  fn0  5069  feq1  5081  foeq1  5154  f1ocnv  5191  mpteqb  5314  eufnfv  5442  tfr0dm  5992  tfrlemiex  6001  tfr1onlemsucfn  6010  tfr1onlemsucaccv  6011  tfr1onlembxssdm  6013  tfr1onlembfn  6014  tfr1onlemex  6017  tfr1onlemaccex  6018  tfr1onlemres  6019