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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 3992 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴) | |
3 | 1, 2 | ssind 4211 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∩ 𝐴)) |
4 | inss2 4208 | . . . . . . . 8 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∩ 𝐴) ⊆ 𝐴) |
6 | 3, 5 | eqssd 3986 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐵 ∩ 𝐴)) |
7 | eleq1 2902 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
8 | ineq1 4183 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
9 | 8 | eqeq2d 2834 | . . . . . . . . . 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 3599 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → ((𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))) |
12 | 11 | expd 418 | . . . . . . 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 509 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) |
15 | df-rex 3146 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) | |
16 | 14, 15 | sylibr 236 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
17 | 16 | adantl 484 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
18 | ssexg 5229 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V) | |
19 | elrest 16703 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) | |
20 | 18, 19 | sylan2 594 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) |
21 | 17, 20 | mpbird 259 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ (𝑋 ↾t 𝐴)) |
22 | 21 | ex 415 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 (class class class)co 7158 ↾t crest 16696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-rest 16698 |
This theorem is referenced by: bj-restv 34388 bj-resta 34389 |
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