Theorem bj-fvsnun2 37400

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7127. (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 5969	. . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3		inss1 4187	. . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
4	2, 3	eqsstri 3978	. . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
5	4	a1i 11	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6		neldifsnd 4747	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
7	5, 6	ssneldd 3934	. 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 37398	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  {csn 4578  ⟨cop 4584  dom cdm 5622   ↾ cres 5624  ‘cfv 6490

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498

This theorem is referenced by: (None)