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Mirrors > Home > MPE Home > Th. List > fvsetsid | Structured version Visualization version GIF version |
Description: The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
Ref | Expression |
---|---|
fvsetsid | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsval 15720 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | |
2 | 1 | 3adant2 1073 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) |
3 | 2 | fveq1d 6105 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
4 | simp2 1055 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝑊) | |
5 | simp3 1056 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
6 | neldifsn 4262 | . . . . 5 ⊢ ¬ 𝑋 ∈ (V ∖ {𝑋}) | |
7 | dmres 5339 | . . . . . . 7 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹) | |
8 | inss1 3795 | . . . . . . 7 ⊢ ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋}) | |
9 | 7, 8 | eqsstri 3598 | . . . . . 6 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋}) |
10 | 9 | sseli 3564 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋})) |
11 | 6, 10 | mto 187 | . . . 4 ⊢ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) |
13 | fsnunfv 6358 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | |
14 | 4, 5, 12, 13 | syl3anc 1318 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
15 | 3, 14 | eqtrd 2644 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 {csn 4125 〈cop 4131 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 sSet csts 15693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-sets 15701 |
This theorem is referenced by: mdetunilem9 20245 |
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