Theorem fneq1 5979

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 5908	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 5324	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2624	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 747	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5891	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5891	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 303	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   dom cdm 5114   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891

This theorem is referenced by:  fneq1d  5981  fneq1i  5985  fn0  6011  feq1  6026  foeq1  6111  f1ocnv  6149  dffn5  6241  mpteqb  6299  fnsnb  6432  fnprb  6472  fntpb  6473  eufnfv  6491  wfrlem1  7414  wfrlem3a  7417  wfrlem15  7429  tfrlem12  7485  mapval2  7887  elixp2  7912  ixpfn  7914  elixpsn  7947  inf3lem6  8530  aceq3lem  8943  dfac4  8945  dfacacn  8963  axcc2lem  9258  axcc3  9260  seqof  12858  ccatvalfn  13365  cshword  13537  0csh0  13539  lmodfopnelem1  18899  rrgsupp  19291  elpt  21375  elptr  21376  ptcmplem3  21858  prdsxmslem2  22334  bnj62  30786  bnj976  30848  bnj66  30930  bnj124  30941  bnj607  30986  bnj873  30994  bnj1234  31081  bnj1463  31123  frrlem1  31780  dssmapf1od  38315  fnchoice  39188  choicefi  39392  axccdom  39416  dfafn5b  41241  cshword2  41437  rngchomffvalALTV  41995