restuni6 - Mathbox for Glauco Siliprandi

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Theorem restuni6 45881

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 2766	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴
4	3	restin 23360	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
5	1, 2, 4	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
6	5	unieqd 4890	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
7		inss2 4193	. . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴)
9	1, 8	restuni4 45880	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴))
10		incom 4165	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
12	6, 9, 11	3eqtrd 2805	1 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ∩ cin 3907 ⊆ wss 3908 ∪ cuni 4877 (class class class)co 7423 ↾_t crest 17498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-rest 17500

This theorem is used by: unirestss 45883

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