Theorem uniopn 14588

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 e. Top /\ A C_ J ) -> U. A e. J )

Proof of Theorem uniopn

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

Step	Hyp	Ref	Expression
1		istopg 14586	. . . . 5 |- ( J e. Top -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
2	1	ibi 176	. . . 4 |- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) )
3	2	simpld 112	. . 3 |- ( J e. Top -> A. x ( x C_ J -> U. x e. J ) )
4		elpw2g 4216	. . . . . . . 8 |- ( J e. Top -> ( A e. ~P J <-> A C_ J ) )
5	4	biimpar 297	. . . . . . 7 |- ( ( J e. Top /\ A C_ J ) -> A e. ~P J )
6		sseq1 3224	. . . . . . . . 9 |- ( x = A -> ( x C_ J <-> A C_ J ) )
7		unieq 3873	. . . . . . . . . 10 |- ( x = A -> U. x = U. A )
8	7	eleq1d 2276	. . . . . . . . 9 |- ( x = A -> ( U. x e. J <-> U. A e. J ) )
9	6, 8	imbi12d 234	. . . . . . . 8 |- ( x = A -> ( ( x C_ J -> U. x e. J ) <-> ( A C_ J -> U. A e. J ) ) )
10	9	spcgv 2867	. . . . . . 7 |- ( A e. ~P J -> ( A. x ( x C_ J -> U. x e. J ) -> ( A C_ J -> U. A e. J ) ) )
11	5, 10	syl 14	. . . . . 6 |- ( ( J e. Top /\ A C_ J ) -> ( A. x ( x C_ J -> U. x e. J ) -> ( A C_ J -> U. A e. J ) ) )
12	11	com23 78	. . . . 5 |- ( ( J e. Top /\ A C_ J ) -> ( A C_ J -> ( A. x ( x C_ J -> U. x e. J ) -> U. A e. J ) ) )
13	12	ex 115	. . . 4 |- ( J e. Top -> ( A C_ J -> ( A C_ J -> ( A. x ( x C_ J -> U. x e. J ) -> U. A e. J ) ) ) )
14	13	pm2.43d 50	. . 3 |- ( J e. Top -> ( A C_ J -> ( A. x ( x C_ J -> U. x e. J ) -> U. A e. J ) ) )
15	3, 14	mpid 42	. 2 |- ( J e. Top -> ( A C_ J -> U. A e. J ) )
16	15	imp 124	1 |- ( ( J e. Top /\ A C_ J ) -> U. A e. J )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2178   A.wral 2486   i^i cin 3173   C_ wss 3174   ~Pcpw 3626   U.cuni 3864   Topctop 14584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-sep 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628  df-uni 3865  df-top 14585

This theorem is referenced by:  iunopn  14589  unopn  14592  0opn  14593  topopn  14595  tgtop  14655  ntropn  14704  neipsm  14741  unimopn  15073  metrest  15093  cnopncntop  15131  cnopn  15132