Theorem fneq1 5304

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 5236	. . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺))
2		dmeq 4827	. . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
3	2	eqeq1d 2186	. . 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 5219	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6		df-fn 5219	. 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 1353  dom cdm 4626  Fun wfun 5210   Fn wfn 5211

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-fun 5218  df-fn 5219

This theorem is referenced by:  fneq1d  5306  fneq1i  5310  fn0  5335  feq1  5348  foeq1  5434  f1ocnv  5474  mpteqb  5606  eufnfv  5747  tfr0dm  6322  tfrlemiex  6331  tfr1onlemsucfn  6340  tfr1onlemsucaccv  6341  tfr1onlembxssdm  6343  tfr1onlembfn  6344  tfr1onlemex  6347  tfr1onlemaccex  6348  tfr1onlemres  6349  mapval2  6677  elixp2  6701  ixpfn  6703  elixpsn  6734  cc2lem  7264  cc3  7266