Theorem inopnt 7601

Description: The intersection of two open sets of a topology is also an open set.

Assertion

Ref	Expression
inopnt	|- ((J e. Top /\ A e. J /\ B e. J) -> (A i^i B) e. J)

Proof of Theorem inopnt

Step	Hyp	Ref	Expression
1		ineq1 2213	. . . . 5 |- (x = A -> (x i^i y) = (A i^i y))
2	1	eleq1d 1543	. . . 4 |- (x = A -> ((x i^i y) e. J <-> (A i^i y) e. J))
3		ineq2 2214	. . . . 5 |- (y = B -> (A i^i y) = (A i^i B))
4	3	eleq1d 1543	. . . 4 |- (y = B -> ((A i^i y) e. J <-> (A i^i B) e. J))
5	2, 4	rcla42v 1883	. . 3 |- ((A e. J /\ B e. J) -> (A.x e. J A.y e. J (x i^i y) e. J -> (A i^i B) e. J))
6		istopg 7598	. . . . 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)))
7	6	ibi 594	. . . 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))
8	7	pm3.27d 325	. . 3 |- (J e. Top -> A.x e. J A.y e. J (x i^i y) e. J)
9	5, 8	syl5com 52	. 2 |- (J e. Top -> ((A e. J /\ B e. J) -> (A i^i B) e. J))
10	9	3impib 833	1 |- ((J e. Top /\ A e. J /\ B e. J) -> (A i^i B) e. J)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648   i^i cin 2049   (_ wss 2050  U.cuni 2507  Topctop 7590

This theorem is referenced by:  topbast 7626  basgen2t 7638  subtop 7643  uncld 7678  innei 7733  qusp 10541

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

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-top 7594

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