Theorem inopn 14797

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 14793	. . . . 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 176	. . . 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 114	. . 3 |- ( J e. Top -> A. x e. J A. y e. J ( x i^i y ) e. J )
4		ineq1 3403	. . . . 5 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
5	4	eleq1d 2300	. . . 4 |- ( x = A -> ( ( x i^i y ) e. J <-> ( A i^i y ) e. J ) )
6		ineq2 3404	. . . . 5 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
7	6	eleq1d 2300	. . . 4 |- ( y = B -> ( ( A i^i y ) e. J <-> ( A i^i B ) e. J ) )
8	5, 7	rspc2v 2924	. . 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 1228	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   i^i cin 3200   C_ wss 3201   U.cuni 3898   Topctop 14791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-top 14792

This theorem is referenced by:  tgclb  14859  topbas  14861  difopn  14902  uncld  14907  ntrin  14918  innei  14957  restopnb  14975  cnptoprest  15033  txcnp  15065  txcnmpt  15067  mopnin  15281