Theorem uniopnt 7599

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

Assertion

Ref	Expression
uniopnt	⊢ ((J ∈ Top ⋀ A ⊆ J) → ∪A ∈ J)

Proof of Theorem uniopnt

Step	Hyp	Ref	Expression
1		istopg 7598	. . . . 5 ⊢ (J ∈ Top → (J ∈ Top ↔ (∀x(x ⊆ J → ∪x ∈ J) ⋀ ∀x ∈ J ∀y ∈ J (x ∩ y) ∈ J)))
2	1	ibi 594	. . . 4 ⊢ (J ∈ Top → (∀x(x ⊆ J → ∪x ∈ J) ⋀ ∀x ∈ J ∀y ∈ J (x ∩ y) ∈ J))
3	2	pm3.26d 321	. . 3 ⊢ (J ∈ Top → ∀x(x ⊆ J → ∪x ∈ J))
4		elpw2g 2732	. . . . . . . 8 ⊢ (J ∈ Top → (A ∈ ℘J ↔ A ⊆ J))
5	4	biimpar 419	. . . . . . 7 ⊢ ((J ∈ Top ⋀ A ⊆ J) → A ∈ ℘J)
6		sseq1 2085	. . . . . . . . 9 ⊢ (x = A → (x ⊆ J ↔ A ⊆ J))
7		unieq 2514	. . . . . . . . . 10 ⊢ (x = A → ∪x = ∪A)
8	7	eleq1d 1543	. . . . . . . . 9 ⊢ (x = A → (∪x ∈ J ↔ ∪A ∈ J))
9	6, 8	imbi12d 628	. . . . . . . 8 ⊢ (x = A → ((x ⊆ J → ∪x ∈ J) ↔ (A ⊆ J → ∪A ∈ J)))
10	9	cla4gv 1865	. . . . . . 7 ⊢ (A ∈ ℘J → (∀x(x ⊆ J → ∪x ∈ J) → (A ⊆ J → ∪A ∈ J)))
11	5, 10	syl 10	. . . . . 6 ⊢ ((J ∈ Top ⋀ A ⊆ J) → (∀x(x ⊆ J → ∪x ∈ J) → (A ⊆ J → ∪A ∈ J)))
12	11	com23 32	. . . . 5 ⊢ ((J ∈ Top ⋀ A ⊆ J) → (A ⊆ J → (∀x(x ⊆ J → ∪x ∈ J) → ∪A ∈ J)))
13	12	ex 373	. . . 4 ⊢ (J ∈ Top → (A ⊆ J → (A ⊆ J → (∀x(x ⊆ J → ∪x ∈ J) → ∪A ∈ J))))
14	13	pm2.43d 65	. . 3 ⊢ (J ∈ Top → (A ⊆ J → (∀x(x ⊆ J → ∪x ∈ J) → ∪A ∈ J)))
15	3, 14	mpid 47	. 2 ⊢ (J ∈ Top → (A ⊆ J → ∪A ∈ J))
16	15	imp 350	1 ⊢ ((J ∈ Top ⋀ A ⊆ J) → ∪A ∈ J)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∀wral 1648   ∩ cin 2049   ⊆ wss 2050  ℘cpw 2405  ∪cuni 2507  Topctop 7590

This theorem is referenced by:  iunopnt 7600  0opnt 7602  topopn 7603  tgval3t 7624  tgtopt 7627  basgen2t 7638  subtop 7643  ntropn 7681  neiint 7716  neips 7724  cncnplem4 7774  qusp 10541  clicls 10593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-pw 2406  df-uni 2508  df-top 7594

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