restuni6 - Mathbox for Glauco Siliprandi

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Theorem restuni6 40838

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 2772	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴
4	3	restin 21490	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
5	1, 2, 4	syl2anc 576	. . 3 ⊢ (𝜑 → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
6	5	unieqd 4718	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
7		inss2 4087	. . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴)
9	1, 8	restuni4 40837	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴))
10		incom 4060	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
12	6, 9, 11	3eqtrd 2812	1 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∩ cin 3822 ⊆ wss 3823 ∪ cuni 4708 (class class class)co 6974 ↾_t crest 16548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-rest 16550

This theorem is referenced by: unirestss 40840