Theorem bj-fvsnun2 35354

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7037. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-fvsnun.un	⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
bj-fvsnun2.ex1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
bj-fvsnun2.ex2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
bj-fvsnun2	⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Proof of Theorem bj-fvsnun2

Step	Hyp	Ref	Expression
1		bj-fvsnun.un	. 2 ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2		dmres 5902	. . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3		inss1 4159	. . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
4	2, 3	eqsstri 3951	. . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
5	4	a1i 11	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6		neldifsnd 4723	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
7	5, 6	ssneldd 3920	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝐹 ↾ (𝐶 ∖ {𝐴})))
8		bj-fvsnun2.ex1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9		bj-fvsnun2.ex2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
10	1, 7, 8, 9	bj-fununsn2 35352	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ⟨cop 4564  dom cdm 5580   ↾ cres 5582  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by: (None)