Theorem fneq1 5381

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 5310	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4897	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2216	. . 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 5293	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5293	. 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 1373   dom cdm 4693   Fun wfun 5284   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-fun 5292  df-fn 5293

This theorem is referenced by:  fneq1d  5383  fneq1i  5387  fn0  5415  feq1  5428  foeq1  5516  f1ocnv  5557  mpteqb  5693  eufnfv  5838  uchoice  6246  tfr0dm  6431  tfrlemiex  6440  tfr1onlemsucfn  6449  tfr1onlemsucaccv  6450  tfr1onlembxssdm  6452  tfr1onlembfn  6453  tfr1onlemex  6456  tfr1onlemaccex  6457  tfr1onlemres  6458  mapval2  6788  elixp2  6812  ixpfn  6814  elixpsn  6845  cc2lem  7413  cc3  7415  lmodfopnelem1  14201