Theorem fneq1d 5474

Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
fneq1d.1	|- ( ph -> F = G )

Assertion

Ref	Expression
fneq1d	|- ( ph -> ( F Fn A <-> G Fn A ) )

Proof of Theorem fneq1d

Step	Hyp	Ref	Expression
1		fneq1d.1	. 2 |- ( ph -> F = G )
2		fneq1 5472	. 2 |- ( F = G -> ( F Fn A <-> G Fn A ) )
3	1, 2	syl 14	1 |- ( ph -> ( F Fn A <-> G Fn A ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   Fn wfn 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-fun 5382  df-fn 5383

This theorem is used by:  fneq12d  5476  f1o00  5679  f1ompt  5862  fmpt2d  5873  f1ocnvd  6295  f1o3d  6301  offval2  6322  ofrfval2  6323  caofinvl  6332  f1od2  6475  cc3  7640  ccatvalfn  11391  swrdlen  11446  plusffng  13744  grpinvfng  13908  grpinvf1o  13934  mulgfng  13986  rng1zrlem  14365  rrgsupp  14683  scaffng  14755  neif  15367  fnmptd  17037