Theorem inopn 11765

Description: The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Assertion

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

Proof of Theorem inopn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istopg 11761	. . . . 5 |- ( J e. Top -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
2	1	ibi 175	. . . 4 |- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) )
3	2	simprd 113	. . 3 |- ( J e. Top -> A. x e. J A. y e. J ( x i^i y ) e. J )
4		ineq1 3197	. . . . 5 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
5	4	eleq1d 2157	. . . 4 |- ( x = A -> ( ( x i^i y ) e. J <-> ( A i^i y ) e. J ) )
6		ineq2 3198	. . . . 5 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
7	6	eleq1d 2157	. . . 4 |- ( y = B -> ( ( A i^i y ) e. J <-> ( A i^i B ) e. J ) )
8	5, 7	rspc2v 2737	. . 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 ) )
9	3, 8	syl5com 29	. 2 |- ( J e. Top -> ( ( A e. J /\ B e. J ) -> ( A i^i B ) e. J ) )
10	9	3impib 1142	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   A.wal 1288   = wceq 1290   e. wcel 1439   A.wral 2360   i^i cin 3001   C_ wss 3002   U.cuni 3661   Topctop 11759

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-in 3008  df-ss 3015  df-pw 3437  df-top 11760

This theorem is referenced by:  tgclb  11828  topbas  11830  difopn  11871  uncld  11876  ntrin  11887  innei  11926