Theorem fneq1 5472

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 5400	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4983	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2247	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 477	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5383	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5383	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   dom cdm 4776   Fun wfun 5374   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:  fneq1d  5474  fneq1i  5478  fn0  5506  feq1  5519  foeq1  5614  f1ocnv  5655  mpteqb  5799  eufnfv  5952  uchoice  6375  tfr0dm  6597  tfrlemiex  6606  tfr1onlemsucfn  6615  tfr1onlemsucaccv  6616  tfr1onlembxssdm  6618  tfr1onlembfn  6619  tfr1onlemex  6622  tfr1onlemaccex  6623  tfr1onlemres  6624  fsetdmprc0  6954  mapval2  6963  elixp2  6988  ixpfn  6990  elixpsn  7021  cc2lem  7638  cc3  7640  lmodfopnelem1  14770