Theorem bj-fvsnun2 37753

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7169. (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 6000	. . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3		inss1 4190	. . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
4	2, 3	eqsstri 3984	. . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
5	4	a1i 11	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6		neldifsnd 4755	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
7	5, 6	ssneldd 3941	. 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 37751	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  {csn 4584  ⟨cop 4590  dom cdm 5649   ↾ cres 5651  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fv 6531

This theorem is referenced by: (None)