Theorem iunopnt 7600

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 2590	. . 3 |- (A.x e. A B e. J -> U_x e. A B = U.{y | E.x e. A y = B})
2	1	adantl 390	. 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 7599	. . 3 |- ((J e. Top /\ {y | E.x e. A y = B} (_ J) -> U.{y | E.x e. A y = B} e. J)
4		uniiunlem 2135	. . . 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 594	. . 3 |- (A.x e. A B e. J -> {y | E.x e. A y = B} (_ J)
6	3, 5	sylan2 453	. 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 1551	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 958   e. wcel 960  {cab 1466  A.wral 1648  E.wrex 1649   (_ wss 2050  U.cuni 2507  U_ciun 2570  Topctop 7590

This theorem is referenced by:  iincld 7676  cncnplem4 7774

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-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-in 2054  df-ss 2056  df-pw 2406  df-uni 2508  df-iun 2572  df-top 7594

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