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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvsnun2 | Structured version Visualization version GIF version |
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 7203. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-fvsnun.un | ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
bj-fvsnun2.ex1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
bj-fvsnun2.ex2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
bj-fvsnun2 | ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fvsnun.un | . 2 ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
2 | dmres 6032 | . . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) | |
3 | inss1 4245 | . . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴}) | |
4 | 2, 3 | eqsstri 4030 | . . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})) |
6 | neldifsnd 4798 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | |
7 | 5, 6 | ssneldd 3998 | . 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 37237 | 1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 {csn 4631 〈cop 4637 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: (None) |
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