Theorem uniopn 12168

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 12166	. . . . 5 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))
2	1	ibi 175	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))
3	2	simpld 111	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽))
4		elpw2g 4081	. . . . . . . 8 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝒫 𝐽 ↔ 𝐴 ⊆ 𝐽))
5	4	biimpar 295	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → 𝐴 ∈ 𝒫 𝐽)
6		sseq1 3120	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐽 ↔ 𝐴 ⊆ 𝐽))
7		unieq 3745	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
8	7	eleq1d 2208	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝐽 ↔ ∪ 𝐴 ∈ 𝐽))
9	6, 8	imbi12d 233	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ↔ (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
10	9	spcgv 2773	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
11	5, 10	syl 14	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽)))
12	11	com23 78	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽)))
13	12	ex 114	. . . 4 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽))))
14	13	pm2.43d 50	. . 3 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) → ∪ 𝐴 ∈ 𝐽)))
15	3, 14	mpid 42	. 2 ⊢ (𝐽 ∈ Top → (𝐴 ⊆ 𝐽 → ∪ 𝐴 ∈ 𝐽))
16	15	imp 123	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∩ cin 3070   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736  Topctop 12164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-top 12165

This theorem is referenced by:  iunopn  12169  unopn  12172  0opn  12173  topopn  12175  tgtop  12237  ntropn  12286  neipsm  12323  unimopn  12655  metrest  12675  cnopncntop  12706