Theorem uniopnt 7598

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 e. Top /\ A (_ J) -> U.A e. J)

Proof of Theorem uniopnt

Step	Hyp	Ref	Expression
1		istopg 7596	. . . . 5 |- (J e. Top -> (J e. Top <-> (A.x(x (_ J -> U.x e. J) /\ A.x e. J A.y e. J (x i^i y) e. J)))
2	1	ibi 592	. . . 4 |- (J e. Top -> (A.x(x (_ J -> U.x e. J) /\ A.x e. J A.y e. J (x i^i y) e. J))
3	2	pm3.26d 321	. . 3 |- (J e. Top -> A.x(x (_ J -> U.x e. J))
4		elpw2g 2727	. . . . . . . 8 |- (J e. Top -> (A e. P~J <-> A (_ J))
5	4	biimpar 417	. . . . . . 7 |- ((J e. Top /\ A (_ J) -> A e. P~J)
6		sseq1 2082	. . . . . . . . 9 |- (x = A -> (x (_ J <-> A (_ J))
7		unieq 2510	. . . . . . . . . 10 |- (x = A -> U.x = U.A)
8	7	eleq1d 1540	. . . . . . . . 9 |- (x = A -> (U.x e. J <-> U.A e. J))
9	6, 8	imbi12d 626	. . . . . . . 8 |- (x = A -> ((x (_ J -> U.x e. J) <-> (A (_ J -> U.A e. J)))
10	9	cla4gv 1862	. . . . . . 7 |- (A e. P~J -> (A.x(x (_ J -> U.x e. J) -> (A (_ J -> U.A e. J)))
11	5, 10	syl 10	. . . . . 6 |- ((J e. Top /\ A (_ J) -> (A.x(x (_ J -> U.x e. J) -> (A (_ J -> U.A e. J)))
12	11	com23 32	. . . . 5 |- ((J e. Top /\ A (_ J) -> (A (_ J -> (A.x(x (_ J -> U.x e. J) -> U.A e. J)))
13	12	ex 373	. . . 4 |- (J e. Top -> (A (_ J -> (A (_ J -> (A.x(x (_ J -> U.x e. J) -> U.A e. J))))
14	13	pm2.43d 65	. . 3 |- (J e. Top -> (A (_ J -> (A.x(x (_ J -> U.x e. J) -> U.A e. J)))
15	3, 14	mpid 47	. 2 |- (J e. Top -> (A (_ J -> U.A e. J))
16	15	imp 350	1 |- ((J e. Top /\ A (_ J) -> U.A e. J)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  P~cpw 2401  U.cuni 2503  Topctop 7588

This theorem is referenced by:  iunopnt 7599  0opnt 7601  topopn 7602  tgval3t 7625  tgtopt 7628  basgen2t 7639  subtop 7646  ntropn 7684  neiint 7719  neips 7727  cncnplem4 7777  qusp 10555  clicls 10622

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504  df-top 7592

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