Theorem fvsnun2 7203

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 7202. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.)

Hypotheses

Ref	Expression
fvsnun.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fvsnun.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fvsnun.3	⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
fvsnun2.4	⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))

Assertion

Ref	Expression
fvsnun2	⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Proof of Theorem fvsnun2

Step	Hyp	Ref	Expression
1		fvsnun.3	. . . . . 6 ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
2	1	reseq1i 5993	. . . . 5 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴}))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})))
4		resundir 6012	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})))
5	4	a1i 11	. . . 4 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))))
6		disjdif 4472	. . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
7		fvsnun.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fvsnun.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		fnsng 6618	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
10	7, 8, 9	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} Fn {𝐴})
11		fnresdisj 6688	. . . . . . . 8 ⊢ ({⟨𝐴, 𝐵⟩} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅))
13	6, 12	mpbii 233	. . . . . 6 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
14		residm 6028	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
16	13, 15	uneq12d 4169	. . . . 5 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))))
17		uncom 4158	. . . . . 6 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅))
19		un0 4394	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
21	16, 18, 20	3eqtrd 2781	. . . 4 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
22	3, 5, 21	3eqtrd 2781	. . 3 ⊢ (𝜑 → (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
23	22	fveq1d 6908	. 2 ⊢ (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
24		fvsnun2.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))
25	24	fvresd 6926	. 2 ⊢ (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷))
26	24	fvresd 6926	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
27	23, 25, 26	3eqtr3d 2785	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950  ∅c0 4333  {csn 4626  ⟨cop 4632   ↾ cres 5687   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  facnn  14314  satfv1lem  35367