Theorem fneq1 5415

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 5344	. . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2		dmeq 4929	. . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
3	2	eqeq1d 2238	. . 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 5327	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6		df-fn 5327	. 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 1395  dom cdm 4723  Fun wfun 5318   Fn wfn 5319

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-fun 5326  df-fn 5327

This theorem is referenced by:  fneq1d  5417  fneq1i  5421  fn0  5449  feq1  5462  foeq1  5552  f1ocnv  5593  mpteqb  5733  eufnfv  5880  uchoice  6295  tfr0dm  6483  tfrlemiex  6492  tfr1onlemsucfn  6501  tfr1onlemsucaccv  6502  tfr1onlembxssdm  6504  tfr1onlembfn  6505  tfr1onlemex  6508  tfr1onlemaccex  6509  tfr1onlemres  6510  mapval2  6842  elixp2  6866  ixpfn  6868  elixpsn  6899  cc2lem  7475  cc3  7477  lmodfopnelem1  14328