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Mirrors > Home > ILE Home > Th. List > fvsetsid | 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 | setsvala 11917 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | |
2 | 1 | fveq1d 5391 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
3 | simp2 967 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝑊) | |
4 | simp3 968 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
5 | neldifsn 3623 | . . . . 5 ⊢ ¬ 𝑋 ∈ (V ∖ {𝑋}) | |
6 | dmres 4810 | . . . . . . 7 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹) | |
7 | inss1 3266 | . . . . . . 7 ⊢ ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋}) | |
8 | 6, 7 | eqsstri 3099 | . . . . . 6 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋}) |
9 | 8 | sseli 3063 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋})) |
10 | 5, 9 | mto 636 | . . . 4 ⊢ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) |
12 | fsnunfv 5589 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | |
13 | 3, 4, 11, 12 | syl3anc 1201 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
14 | 2, 13 | eqtrd 2150 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 947 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∖ cdif 3038 ∪ cun 3039 ∩ cin 3040 {csn 3497 〈cop 3500 dom cdm 4509 ↾ cres 4511 ‘cfv 5093 (class class class)co 5742 sSet csts 11884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sets 11893 |
This theorem is referenced by: (None) |
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