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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 11993 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | |
2 | 1 | fveq1d 5423 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
3 | simp2 982 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝑊) | |
4 | simp3 983 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
5 | neldifsn 3653 | . . . . 5 ⊢ ¬ 𝑋 ∈ (V ∖ {𝑋}) | |
6 | dmres 4840 | . . . . . . 7 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹) | |
7 | inss1 3296 | . . . . . . 7 ⊢ ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋}) | |
8 | 6, 7 | eqsstri 3129 | . . . . . 6 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋}) |
9 | 8 | sseli 3093 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋})) |
10 | 5, 9 | mto 651 | . . . 4 ⊢ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) |
11 | 10 | a1i 9 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) |
12 | fsnunfv 5621 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | |
13 | 3, 4, 11, 12 | syl3anc 1216 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
14 | 2, 13 | eqtrd 2172 | 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 962 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ∪ cun 3069 ∩ cin 3070 {csn 3527 〈cop 3530 dom cdm 4539 ↾ cres 4541 ‘cfv 5123 (class class class)co 5774 sSet csts 11960 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sets 11969 |
This theorem is referenced by: (None) |
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