Theorem fvsnun1 6946

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6947. (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 5851	. . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴})
3		resundir 5870	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}))
4		incom 4180	. . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴}))
5		disjdif 4423	. . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
6	4, 5	eqtri 2846	. . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅
7		resdisj 6028	. . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅)
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅
9	8	uneq2i 4138	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅)
10		un0 4346	. . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
11	9, 10	eqtri 2846	. . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
12	3, 11	eqtri 2846	. . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
13	2, 12	eqtri 2846	. . 3 ⊢ (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴})
14	13	fveq1i 6673	. 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴)
15		fvsnun.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16		snidg 4601	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
17	15, 16	syl 17	. . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
18	17	fvresd 6692	. 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴))
19	17	fvresd 6692	. . 3 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
20		fvsnun.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
21		fvsng 6944	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
22	15, 20, 21	syl2anc 586	. . 3 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
23	19, 22	eqtrd 2858	. 2 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵)
24	14, 18, 23	3eqtr3a 2882	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937  ∅c0 4293  {csn 4569  ⟨cop 4575   ↾ cres 5559  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by:  fac0  13639  ruclem4  15589  satfv1lem  32611