Theorem iunopnt 7599

Description: The indexed union of a subset of a topology is an open set.

Assertion

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

Distinct variable groups:   x,A   x,J

Proof of Theorem iunopnt

Step	Hyp	Ref	Expression
1		dfiun2g 2586	. . 3 |- (A.x e. A B e. J -> U_x e. A B = U.{y | E.x e. A y = B})
2	1	adantl 388	. 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		uniopnt 7598	. . 3 |- ((J e. Top /\ {y | E.x e. A y = B} (_ J) -> U.{y | E.x e. A y = B} e. J)
4		uniiunlem 2132	. . . 4 |- (A.x e. A B e. J -> (A.x e. A B e. J <-> {y | E.x e. A y = B} (_ J))
5	4	ibi 592	. . 3 |- (A.x e. A B e. J -> {y | E.x e. A y = B} (_ J)
6	3, 5	sylan2 451	. 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 1548	1 |- ((J e. Top /\ A.x e. A B e. J) -> U_x e. A B e. J)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646   (_ wss 2047  U.cuni 2503  U_ciun 2566  Topctop 7588

This theorem is referenced by:  iincld 7679  cncnplem4 7777

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-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504  df-iun 2568  df-top 7592

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