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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 4019 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | ssind 4249 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∩ 𝐴)) |
4 | inss2 4246 | . . . . . . . 8 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∩ 𝐴) ⊆ 𝐴) |
6 | 3, 5 | eqssd 4013 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐵 ∩ 𝐴)) |
7 | eleq1 2827 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
8 | ineq1 4221 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
9 | 8 | eqeq2d 2746 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 = (𝑦 ∩ 𝐴) ↔ 𝐴 = (𝐵 ∩ 𝐴))) |
10 | 7, 9 | anbi12d 632 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)))) |
11 | 10 | spcegv 3597 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → ((𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))) |
12 | 11 | expd 415 | . . . . . . 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 506 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) |
15 | df-rex 3069 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) | |
16 | 14, 15 | sylibr 234 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
17 | 16 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
18 | ssexg 5329 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V) | |
19 | elrest 17474 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) | |
20 | 18, 19 | sylan2 593 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) |
21 | 17, 20 | mpbird 257 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ (𝑋 ↾t 𝐴)) |
22 | 21 | ex 412 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 (class class class)co 7431 ↾t crest 17467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rest 17469 |
This theorem is referenced by: bj-restv 37078 bj-resta 37079 |
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