Theorem fneq1 5444

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 5372	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4956	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2241	. . 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 5355	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5355	. 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 4749   Fun wfun 5346   Fn wfn 5347

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-fun 5354  df-fn 5355

This theorem is referenced by:  fneq1d  5446  fneq1i  5450  fn0  5478  feq1  5491  foeq1  5586  f1ocnv  5627  mpteqb  5768  eufnfv  5917  uchoice  6331  tfr0dm  6553  tfrlemiex  6562  tfr1onlemsucfn  6571  tfr1onlemsucaccv  6572  tfr1onlembxssdm  6574  tfr1onlembfn  6575  tfr1onlemex  6578  tfr1onlemaccex  6579  tfr1onlemres  6580  mapval2  6912  elixp2  6937  ixpfn  6939  elixpsn  6970  cc2lem  7580  cc3  7582  lmodfopnelem1  14472