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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 3910 | . . . . . . . 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 3904 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (𝐵 ∩ 𝐴)) |
7 | eleq1 2821 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋)) | |
8 | ineq1 4106 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (𝑦 ∩ 𝐴) = (𝐵 ∩ 𝐴)) | |
9 | 8 | eqeq2d 2750 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 = (𝑦 ∩ 𝐴) ↔ 𝐴 = (𝐵 ∩ 𝐴))) |
10 | 7, 9 | anbi12d 634 | . . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)) ↔ (𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)))) |
11 | 10 | spcegv 3504 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑋 → ((𝐵 ∈ 𝑋 ∧ 𝐴 = (𝐵 ∩ 𝐴)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴)))) |
12 | 11 | expd 419 | . . . . . . 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 510 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) |
15 | df-rex 3060 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ 𝐴 = (𝑦 ∩ 𝐴))) | |
16 | 14, 15 | sylibr 237 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
17 | 16 | adantl 485 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴)) |
18 | ssexg 5201 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V) | |
19 | elrest 16817 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) | |
20 | 18, 19 | sylan2 596 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝑋 𝐴 = (𝑦 ∩ 𝐴))) |
21 | 17, 20 | mpbird 260 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ (𝑋 ↾t 𝐴)) |
22 | 21 | ex 416 | 1 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ (𝑋 ↾t 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 ∃wrex 3055 Vcvv 3400 ∩ cin 3852 ⊆ wss 3853 (class class class)co 7183 ↾t crest 16810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7186 df-oprab 7187 df-mpo 7188 df-rest 16812 |
This theorem is referenced by: bj-restv 34920 bj-resta 34921 |
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