Theorem inopn 12641

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 12637	. . . . 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 3316	. . . . 5 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
5	4	eleq1d 2235	. . . 4 |- ( x = A -> ( ( x i^i y ) e. J <-> ( A i^i y ) e. J ) )
6		ineq2 3317	. . . . 5 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
7	6	eleq1d 2235	. . . 4 |- ( y = B -> ( ( A i^i y ) e. J <-> ( A i^i B ) e. J ) )
8	5, 7	rspc2v 2843	. . 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 1191	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   i^i cin 3115   C_ wss 3116   U.cuni 3789   Topctop 12635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-top 12636

This theorem is referenced by:  tgclb  12705  topbas  12707  difopn  12748  uncld  12753  ntrin  12764  innei  12803  restopnb  12821  cnptoprest  12879  txcnp  12911  txcnmpt  12913  mopnin  13127