Theorem uniopn 22918

Description: The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

Assertion

Ref	Expression
uniopn	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Proof of Theorem uniopn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istopg 22916	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
2	1	ibi 267	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
3	2	simpld 494	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))
4		elpw2g 5338	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽 ↔ 𝐴 ⊆ 𝐽))
5	4	biimpar 477	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → 𝐴 ∈ 𝒫 𝐽)
6		sseq1 4020	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐽 ↔ 𝐴 ⊆ 𝐽))
7		unieq 4922	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
8	7	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝐽 ↔ ∪ 𝐴 ∈ 𝐽))
9	6, 8	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
10	9	spcgv 3595	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
11	5, 10	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
12	11	com23 86	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽)))
13	12	ex 412	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽))))
14	13	pm2.43d 53	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽)))
15	3, 14	mpid 44	. 2 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽))
16	15	imp 406	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1534   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911  Topctop 22914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-in 3969  df-ss 3979  df-pw 4606  df-uni 4912  df-top 22915

This theorem is referenced by:  iunopn  22919  unopn  22924  0opn  22925  topopn  22927  tgtop  22995  ntropn  23072  toponmre  23116  neips  23136  txcmplem1  23664  unimopn  24524  metrest  24552  cnopn  24822  locfinreflem  33800  cvmscld  35257  mblfinlem3  37645  mblfinlem4  37646  ismblfin  37647  topclat  48786  toplatlub  48788