Theorem inopnt 7600

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 2210	. . . . 5 |- (x = A -> (x i^i y) = (A i^i y))
2	1	eleq1d 1540	. . . 4 |- (x = A -> ((x i^i y) e. J <-> (A i^i y) e. J))
3		ineq2 2211	. . . . 5 |- (y = B -> (A i^i y) = (A i^i B))
4	3	eleq1d 1540	. . . 4 |- (y = B -> ((A i^i y) e. J <-> (A i^i B) e. J))
5	2, 4	rcla42v 1880	. . 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 7596	. . . . 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 592	. . . 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 831	1 |- ((J e. Top /\ A e. J /\ B e. J) -> (A i^i B) e. J)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  U.cuni 2503  Topctop 7588

This theorem is referenced by:  topbast 7627  basgen2t 7639  subtop 7646  uncld 7681  innei 7736  qusp 10555

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

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-top 7592

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