restuni6 - Mathbox for Glauco Siliprandi

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Theorem restuni6 45024

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 2740	. . . . 5 ⊢ ∪ 𝐴 = ∪ 𝐴
4	3	restin 23195	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
5	1, 2, 4	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐴 ↾_t 𝐵) = (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
6	5	unieqd 4944	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)))
7		inss2 4259	. . . 4 ⊢ (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) ⊆ ∪ 𝐴)
9	1, 8	restuni4 45023	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾_t (𝐵 ∩ ∪ 𝐴)) = (𝐵 ∩ ∪ 𝐴))
10		incom 4230	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
12	6, 9, 11	3eqtrd 2784	1 ⊢ (𝜑 → ∪ (𝐴 ↾_t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ⊆ wss 3976 ∪ cuni 4931 (class class class)co 7448 ↾_t crest 17480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-rest 17482

This theorem is referenced by: unirestss 45026