Theorem uniopn 22036

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 22034	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
2	1	ibi 266	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
3	2	simpld 495	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))
4		elpw2g 5272	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽 ↔ 𝐴 ⊆ 𝐽))
5	4	biimpar 478	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → 𝐴 ∈ 𝒫 𝐽)
6		sseq1 3951	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐽 ↔ 𝐴 ⊆ 𝐽))
7		unieq 4856	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
8	7	eleq1d 2825	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝐽 ↔ ∪ 𝐴 ∈ 𝐽))
9	6, 8	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
10	9	spcgv 3534	. . . . . . 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 413	. . . 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 407	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1540   = wceq 1542   ∈ wcel 2110  ∀wral 3066   ∩ cin 3891   ⊆ wss 3892  𝒫 cpw 4539  ∪ cuni 4845  Topctop 22032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-in 3899  df-ss 3909  df-pw 4541  df-uni 4846  df-top 22033

This theorem is referenced by:  iunopn  22037  unopn  22042  0opn  22043  topopn  22045  tgtop  22113  ntropn  22190  toponmre  22234  neips  22254  txcmplem1  22782  unimopn  23642  metrest  23670  cnopn  23940  locfinreflem  31778  cvmscld  33223  mblfinlem3  35804  mblfinlem4  35805  ismblfin  35806  topclat  46245  toplatlub  46247