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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restb | Structured version Visualization version GIF version |
Description: An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restb | ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | ssidd 3915 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | ssind 4133 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∩ 𝐴)) |
4 | inss2 4130 | . . . . . . . 8 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∩ 𝐴) ⊆ 𝐴) |
6 | 3, 5 | eqssd 3910 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐵 ∩ 𝐴)) |
7 | eleq1 2870 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
8 | ineq1 4105 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
9 | 8 | eqeq2d 2805 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 = (𝑦 ∩ 𝐴) ↔ 𝐴 = (𝐵 ∩ 𝐴))) |
10 | 7, 9 | anbi12d 630 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)))) |
11 | 10 | spcegv 3540 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → ((𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))) |
12 | 11 | expd 416 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 = (𝐵 ∩ 𝐴) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))))) |
13 | 12 | pm2.43i 52 | . . . . . 6 ⊢ (𝐵 ∈ 𝑋 → (𝐴 = (𝐵 ∩ 𝐴) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))) |
14 | 6, 13 | mpan9 507 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) |
15 | df-rex 3111 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) | |
16 | 14, 15 | sylibr 235 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
17 | 16 | adantl 482 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
18 | ssexg 5123 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V) | |
19 | elrest 16535 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) | |
20 | 18, 19 | sylan2 592 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) |
21 | 17, 20 | mpbird 258 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ (𝑋 ↾t 𝐴)) |
22 | 21 | ex 413 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∃wrex 3106 Vcvv 3437 ∩ cin 3862 ⊆ wss 3863 (class class class)co 7021 ↾t crest 16528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-ov 7024 df-oprab 7025 df-mpo 7026 df-rest 16530 |
This theorem is referenced by: bj-restv 34011 bj-resta 34012 |
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