Theorem fneq1 5346

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 5278	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4866	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2205	. . 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 5261	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5261	. 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 1364   dom cdm 4663   Fun wfun 5252   Fn wfn 5253

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-fun 5260  df-fn 5261

This theorem is referenced by:  fneq1d  5348  fneq1i  5352  fn0  5377  feq1  5390  foeq1  5476  f1ocnv  5517  mpteqb  5652  eufnfv  5793  uchoice  6195  tfr0dm  6380  tfrlemiex  6389  tfr1onlemsucfn  6398  tfr1onlemsucaccv  6399  tfr1onlembxssdm  6401  tfr1onlembfn  6402  tfr1onlemex  6405  tfr1onlemaccex  6406  tfr1onlemres  6407  mapval2  6737  elixp2  6761  ixpfn  6763  elixpsn  6794  cc2lem  7333  cc3  7335  lmodfopnelem1  13880