Theorem inopn 12795

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 12791	. . . . 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 3321	. . . . 5 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
5	4	eleq1d 2239	. . . 4 |- ( x = A -> ( ( x i^i y ) e. J <-> ( A i^i y ) e. J ) )
6		ineq2 3322	. . . . 5 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
7	6	eleq1d 2239	. . . 4 |- ( y = B -> ( ( A i^i y ) e. J <-> ( A i^i B ) e. J ) )
8	5, 7	rspc2v 2847	. . 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 1196	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   C_ wss 3121   U.cuni 3796   Topctop 12789

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-top 12790

This theorem is referenced by:  tgclb  12859  topbas  12861  difopn  12902  uncld  12907  ntrin  12918  innei  12957  restopnb  12975  cnptoprest  13033  txcnp  13065  txcnmpt  13067  mopnin  13281