Theorem fneq1 5446

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 5374	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4958	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2243	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5357	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5357	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   dom cdm 4751   Fun wfun 5348   Fn wfn 5349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-fun 5356  df-fn 5357

This theorem is referenced by:  fneq1d  5448  fneq1i  5452  fn0  5480  feq1  5493  foeq1  5588  f1ocnv  5629  mpteqb  5770  eufnfv  5919  uchoice  6333  tfr0dm  6555  tfrlemiex  6564  tfr1onlemsucfn  6573  tfr1onlemsucaccv  6574  tfr1onlembxssdm  6576  tfr1onlembfn  6577  tfr1onlemex  6580  tfr1onlemaccex  6581  tfr1onlemres  6582  mapval2  6914  elixp2  6939  ixpfn  6941  elixpsn  6972  cc2lem  7582  cc3  7584  lmodfopnelem1  14489