Theorem fvsnun2 6934

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6933. (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 5817	. . . . 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 5836	. . . . 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 4366	. . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅
7		fvsnun.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fvsnun.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9		fnsng 6385	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
10	7, 8, 9	syl2anc 588	. . . . . . . 8 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} Fn {𝐴})
11		fnresdisj 6448	. . . . . . . 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 236	. . . . . 6 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) = ∅)
14		residm 5854	. . . . . . 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 4070	. . . . 5 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))))
17		uncom 4059	. . . . . 6 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅)
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅))
19		un0 4285	. . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴}))
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
21	16, 18, 20	3eqtrd 2798	. . . 4 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
22	3, 5, 21	3eqtrd 2798	. . 3 ⊢ (𝜑 → (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})))
23	22	fveq1d 6658	. 2 ⊢ (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
24		fvsnun2.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))
25	24	fvresd 6676	. 2 ⊢ (𝜑 → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷))
26	24	fvresd 6676	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
27	23, 25, 26	3eqtr3d 2802	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858  ∅c0 4226  {csn 4520  ⟨cop 4526   ↾ cres 5524   Fn wfn 6328  ‘cfv 6333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-uni 4797  df-br 5031  df-opab 5093  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-res 5534  df-iota 6292  df-fun 6335  df-fn 6336  df-fv 6341

This theorem is referenced by:  facnn  13675  satfv1lem  32830