Theorem fiinbas 8885

Description: If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
fiinbas	|- ((B e. C /\ A.x e. B A.y e. B (x i^i y) e. B) -> B e. Bases)

Distinct variable groups:   x,B,y   x,C,y

Proof of Theorem fiinbas

Step	Hyp	Ref	Expression
1		ssid 2634	. . . . . . . 8 |- (x i^i y) C_ (x i^i y)
2		eleq2 1958	. . . . . . . . . 10 |- (w = (x i^i y) -> (z e. w <-> z e. (x i^i y)))
3		sseq1 2637	. . . . . . . . . 10 |- (w = (x i^i y) -> (w C_ (x i^i y) <-> (x i^i y) C_ (x i^i y)))
4	2, 3	anbi12d 690	. . . . . . . . 9 |- (w = (x i^i y) -> ((z e. w /\ w C_ (x i^i y)) <-> (z e. (x i^i y) /\ (x i^i y) C_ (x i^i y))))
5	4	rcla4ev 2381	. . . . . . . 8 |- (((x i^i y) e. B /\ (z e. (x i^i y) /\ (x i^i y) C_ (x i^i y))) -> E.w e. B (z e. w /\ w C_ (x i^i y)))
6	1, 5	mpanr2 776	. . . . . . 7 |- (((x i^i y) e. B /\ z e. (x i^i y)) -> E.w e. B (z e. w /\ w C_ (x i^i y)))
7	6	r19.21aiva 2176	. . . . . 6 |- ((x i^i y) e. B -> A.z e. (x i^i y)E.w e. B (z e. w /\ w C_ (x i^i y)))
8	7	a1i 8	. . . . 5 |- (B e. C -> ((x i^i y) e. B -> A.z e. (x i^i y)E.w e. B (z e. w /\ w C_ (x i^i y))))
9	8	ralimdv 2172	. . . 4 |- (B e. C -> (A.y e. B (x i^i y) e. B -> A.y e. B A.z e. (x i^i y)E.w e. B (z e. w /\ w C_ (x i^i y))))
10	9	ralimdv 2172	. . 3 |- (B e. C -> (A.x e. B A.y e. B (x i^i y) e. B -> A.x e. B A.y e. B A.z e. (x i^i y)E.w e. B (z e. w /\ w C_ (x i^i y))))
11		isbasis2g 8881	. . 3 |- (B e. C -> (B e. Bases <-> A.x e. B A.y e. B A.z e. (x i^i y)E.w e. B (z e. w /\ w C_ (x i^i y))))
12	10, 11	sylibrd 221	. 2 |- (B e. C -> (A.x e. B A.y e. B (x i^i y) e. B -> B e. Bases))
13	12	imp 377	1 |- ((B e. C /\ A.x e. B A.y e. B (x i^i y) e. B) -> B e. Bases)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  Basesctb 8859

This theorem is referenced by:  txbas 8933

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-bases 8863

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