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Mirrors > Home > MPE Home > Th. List > fvsnun1 | 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, 25-Feb-2023.) |
Ref | Expression |
---|---|
fvsnun.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fvsnun.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fvsnun.3 | ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
Ref | Expression |
---|---|
fvsnun1 | ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsnun.3 | . . . . 5 ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) | |
2 | 1 | reseq1i 5996 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 6015 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | disjdifr 4479 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ | |
5 | resdisj 6191 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
7 | 6 | uneq2i 4175 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) |
8 | un0 4400 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
9 | 7, 8 | eqtri 2763 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
10 | 3, 9 | eqtri 2763 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
11 | 2, 10 | eqtri 2763 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
12 | 11 | fveq1i 6908 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) |
13 | fvsnun.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | snidg 4665 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
16 | 15 | fvresd 6927 | . 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) |
17 | 15 | fvresd 6927 | . . 3 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
18 | fvsnun.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
19 | fvsng 7200 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
20 | 13, 18, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
21 | 17, 20 | eqtrd 2775 | . 2 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵) |
22 | 12, 16, 21 | 3eqtr3a 2799 | 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 ∅c0 4339 {csn 4631 〈cop 4637 ↾ 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-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-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-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: fac0 14312 ruclem4 16267 satfv1lem 35347 |
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