Theorem bj-fvsnun2 37237

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7139. (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 5972	. . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3		inss1 4196	. . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
4	2, 3	eqsstri 3990	. . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
5	4	a1i 11	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6		neldifsnd 4753	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
7	5, 6	ssneldd 3946	. 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 37235	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  {csn 4585  ⟨cop 4591  dom cdm 5631   ↾ cres 5633  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fv 6507

This theorem is referenced by: (None)