Theorem iunopn 20703

Description: The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Assertion

Ref	Expression
iunopn	|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )

Distinct variable groups:   x, A   x, J

Allowed substitution hint:   B( x)

Proof of Theorem iunopn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 4552	. . 3 |- ( A. x e. A B e. J -> U_ x e. A B = U. { y | E. x e. A y = B } )
2	1	adantl 482	. 2 |- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B = U. { y | E. x e. A y = B } )
3		uniiunlem 3691	. . . 4 |- ( A. x e. A B e. J -> ( A. x e. A B e. J <-> { y | E. x e. A y = B } C_ J ) )
4	3	ibi 256	. . 3 |- ( A. x e. A B e. J -> { y | E. x e. A y = B } C_ J )
5		uniopn 20702	. . 3 |- ( ( J e. Top /\ { y | E. x e. A y = B } C_ J ) -> U. { y | E. x e. A y = B } e. J )
6	4, 5	sylan2 491	. 2 |- ( ( J e. Top /\ A. x e. A B e. J ) -> U. { y | E. x e. A y = B } e. J )
7	2, 6	eqeltrd 2701	1 |- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   U_ciun 4520   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-iun 4522  df-top 20699

This theorem is referenced by:  iincld  20843  tgcn  21056  kgentopon  21341  xkococnlem  21462  qtoptop2  21502  zcld  22616  metnrmlem2  22663  cnambfre  33458