Theorem fneq1 6438

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 6369	. . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2		dmeq 5766	. . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
3	2	eqeq1d 2823	. . 3 ⊢ (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴))
4	1, 3	anbi12d 632	. 2 ⊢ (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)))
5		df-fn 6352	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6		df-fn 6352	. 2 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
7	4, 5, 6	3bitr4g 316	1 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  dom cdm 5549  Fun wfun 6343   Fn wfn 6344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-fun 6351  df-fn 6352

This theorem is referenced by:  fneq1d  6440  fneq1i  6444  fn0  6473  feq1  6489  foeq1  6580  f1ocnv  6621  dffn5  6718  mpteqb  6781  fnsnb  6922  fnprb  6965  fntpb  6966  eufnfv  6985  wfrlem1  7948  wfrlem3a  7951  wfrlem15  7963  tfrlem12  8019  mapval2  8430  elixp2  8459  ixpfn  8461  elixpsn  8495  inf3lem6  9090  aceq3lem  9540  dfac4  9542  dfacacn  9561  axcc2lem  9852  axcc3  9854  seqof  13421  ccatvalfn  13929  cshword  14147  0csh0  14149  lmodfopnelem1  19664  rrgsupp  20058  elpt  22174  elptr  22175  ptcmplem3  22656  prdsxmslem2  23133  tgjustr  26254  bnj62  31985  bnj976  32044  bnj66  32127  bnj124  32138  bnj607  32183  bnj873  32191  bnj1234  32280  bnj1463  32322  frrlem1  33118  frrlem13  33130  fnsnbt  39113  dssmapf1od  40360  fnchoice  41279  choicefi  41456  axccdom  41480  dfafn5b  43354  rngchomffvalALTV  44260