restuni6 - Mathbox for Glauco Siliprandi

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Theorem restuni6 42344

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 2737	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴
4	3	restin 22063	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
5	1, 2, 4	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
6	5	unieqd 4833	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
7		inss2 4144	. . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴)
9	1, 8	restuni4 42343	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴))
10		incom 4115	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
12	6, 9, 11	3eqtrd 2781	1 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 ∪ cuni 4819 (class class class)co 7213 ↾_t crest 16925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-rest 16927

This theorem is referenced by: unirestss 42346