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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 12029 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | |
2 | 1 | fveq1d 5431 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
3 | simp2 983 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝑊) | |
4 | simp3 984 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
5 | neldifsn 3661 | . . . . 5 ⊢ ¬ 𝑋 ∈ (V ∖ {𝑋}) | |
6 | dmres 4848 | . . . . . . 7 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹) | |
7 | inss1 3301 | . . . . . . 7 ⊢ ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋}) | |
8 | 6, 7 | eqsstri 3134 | . . . . . 6 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋}) |
9 | 8 | sseli 3098 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋})) |
10 | 5, 9 | mto 652 | . . . 4 ⊢ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) |
12 | fsnunfv 5629 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | |
13 | 3, 4, 11, 12 | syl3anc 1217 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
14 | 2, 13 | eqtrd 2173 | 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 963 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∖ cdif 3073 ∪ cun 3074 ∩ cin 3075 {csn 3532 〈cop 3535 dom cdm 4547 ↾ cres 4549 ‘cfv 5131 (class class class)co 5782 sSet csts 11996 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sets 12005 |
This theorem is referenced by: (None) |
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