Theorem inopn 14446

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 14442	. . . . 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 3366	. . . . 5 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
5	4	eleq1d 2273	. . . 4 |- ( x = A -> ( ( x i^i y ) e. J <-> ( A i^i y ) e. J ) )
6		ineq2 3367	. . . . 5 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
7	6	eleq1d 2273	. . . 4 |- ( y = B -> ( ( A i^i y ) e. J <-> ( A i^i B ) e. J ) )
8	5, 7	rspc2v 2889	. . 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 1203	1 |- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1370   = wceq 1372   e. wcel 2175   A.wral 2483   i^i cin 3164   C_ wss 3165   U.cuni 3849   Topctop 14440

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-sep 4161

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-top 14441

This theorem is referenced by:  tgclb  14508  topbas  14510  difopn  14551  uncld  14556  ntrin  14567  innei  14606  restopnb  14624  cnptoprest  14682  txcnp  14714  txcnmpt  14716  mopnin  14930