Theorem fvsnun1 7197

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7198. (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})))

Assertion

Ref	Expression
fvsnun1	⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Proof of Theorem fvsnun1

Step	Hyp	Ref	Expression
1		fvsnun.3	. . . . 5 ⊢ 𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))
2	1	reseq1i 5971	. . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴})
3		resundir 5990	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}))
4		disjdifr 4430	. . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅
5		resdisj 6167	. . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅
7	6	uneq2i 4167	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅)
8		un0 4347	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
9	7, 8	eqtri 2786	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
10	3, 9	eqtri 2786	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
11	2, 10	eqtri 2786	. . 3 ⊢ (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
12	11	fveq1i 6897	. 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴)
13		fvsnun.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14		snidg 4621	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
15	13, 14	syl 18	. . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
16	15	fvresd 6916	. 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴))
17	15	fvresd 6916	. . 3 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
18		fvsnun.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
19		fvsng 7195	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
20	13, 18, 19	syl2anc 596	. . 3 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
21	17, 20	eqtrd 2798	. 2 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵)
22	12, 16, 21	3eqtr3a 2822	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838   ↾ cres 5657  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6501  df-fun 6549  df-fv 6555

This theorem is used by:  fac0  14433  ruclem4  16417  satfv1lem  36251