Theorem bj-fvsnun2 37251

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7160. (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 5986	. . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹)
3		inss1 4203	. . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴})
4	2, 3	eqsstri 3996	. . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})
5	4	a1i 11	. . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}))
6		neldifsnd 4760	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴}))
7	5, 6	ssneldd 3952	. 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 37249	1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ⟨cop 4598  dom cdm 5641   ↾ cres 5643  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522

This theorem is referenced by: (None)