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Theorem fneq1 6658
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
fneq1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))

Proof of Theorem fneq1
StepHypRef Expression
1 funeq 6585 . . 3 (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2 dmeq 5913 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
32eqeq1d 2738 . . 3 (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴))
41, 3anbi12d 632 . 2 (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)))
5 df-fn 6563 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6 df-fn 6563 . 2 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
74, 5, 63bitr4g 314 1 (𝐹 = 𝐺 → (𝐹 Fn 𝐴𝐺 Fn 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  dom cdm 5684  Fun wfun 6554   Fn wfn 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-fun 6562  df-fn 6563
This theorem is referenced by:  fneq1d  6660  fneq1i  6664  fn0  6698  feq1  6715  foeq1  6815  f1ocnv  6859  dffn5  6966  mpteqb  7034  fnsnbg  7185  fnsnbOLD  7187  fnprb  7229  fntpb  7230  eufnfv  7250  frrlem1  8312  frrlem13  8324  wfrlem1OLD  8349  wfrlem3OLDa  8352  wfrlem15OLD  8364  tfrlem12  8430  fsetdmprc0  8896  mapval2  8913  elixp2  8942  ixpfn  8944  elixpsn  8978  inf3lem6  9674  ssttrcl  9756  ttrcltr  9757  ttrclss  9761  ttrclselem2  9767  aceq3lem  10161  dfac4  10163  dfacacn  10183  axcc2lem  10477  axcc3  10479  seqof  14101  ccatvalfn  14620  cshword  14830  0csh0  14832  rrgsupp  20702  lmodfopnelem1  20897  elpt  23581  elptr  23582  ptcmplem3  24063  prdsxmslem2  24543  tgjustr  28483  bnj62  34735  bnj976  34792  bnj66  34875  bnj124  34886  bnj607  34931  bnj873  34939  bnj1234  35028  bnj1463  35070  fineqvac  35112  gblacfnacd  35114  eqresfnbd  42273  dssmapf1od  44039  fnchoice  45039  choicefi  45210  axccdom  45232  dfafn5b  47178  rngchomffvalALTV  48199  functhinclem1  49118
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