restuni6 - Mathbox for Glauco Siliprandi

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Theorem restuni6 42671

Description: The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
restuni6.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
restuni6.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

**Assertion**
Ref	Expression
restuni6	⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

**Proof of Theorem restuni6**
Step	Hyp	Ref	Expression
1		restuni6.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		restuni6.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		eqid 2738	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴
4	3	restin 22317	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
5	1, 2, 4	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
6	5	unieqd 4853	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
7		inss2 4163	. . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴)
9	1, 8	restuni4 42670	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴))
10		incom 4135	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
12	6, 9, 11	3eqtrd 2782	1 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ∪ cuni 4839 (class class class)co 7275 ↾_t crest 17131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-rest 17133

This theorem is referenced by: unirestss 42673