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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 6922. (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 5814 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 5833 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | incom 4128 | . . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴})) | |
5 | disjdif 4379 | . . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
6 | 4, 5 | eqtri 2821 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ |
7 | resdisj 5993 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
9 | 8 | uneq2i 4087 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) |
10 | un0 4298 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
11 | 9, 10 | eqtri 2821 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
12 | 3, 11 | eqtri 2821 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
13 | 2, 12 | eqtri 2821 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
14 | 13 | fveq1i 6646 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) |
15 | fvsnun.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
16 | snidg 4559 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
18 | 17 | fvresd 6665 | . 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) |
19 | 17 | fvresd 6665 | . . 3 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) |
20 | fvsnun.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
21 | fvsng 6919 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
22 | 15, 20, 21 | syl2anc 587 | . . 3 ⊢ (𝜑 → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) |
23 | 19, 22 | eqtrd 2833 | . 2 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵) |
24 | 14, 18, 23 | 3eqtr3a 2857 | 1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 〈cop 4531 ↾ cres 5521 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: fac0 13632 ruclem4 15579 satfv1lem 32722 |
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