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Mirrors > Home > MPE Home > Th. List > Mathboxes > restuni6 | Structured version Visualization version GIF version |
Description: The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
restuni6.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
restuni6.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
restuni6 | ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni6.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | restuni6.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | eqid 2821 | . . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
4 | 3 | restin 21773 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾t 𝐵) = (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
5 | 1, 2, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐴 ↾t 𝐵) = (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
6 | 5 | unieqd 4851 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = ∪ (𝐴 ↾t (𝐵 ∩ ∪ 𝐴))) |
7 | inss2 4205 | . . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴 | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴) |
9 | 1, 8 | restuni4 41385 | . 2 ⊢ (𝜑 → ∪ (𝐴 ↾t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴)) |
10 | incom 4177 | . . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)) |
12 | 6, 9, 11 | 3eqtrd 2860 | 1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4837 (class class class)co 7155 ↾t crest 16693 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-rest 16695 |
This theorem is referenced by: unirestss 41388 |
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