Theorem filint 10473

Description: A filter is closed under taking intersections.

Assertion

Ref	Expression
filint	|- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)

Proof of Theorem filint

Step	Hyp	Ref	Expression
1		eqid 1473	. . . . . . . . 9 |- U.F = U.F
2	1	isfil 10469	. . . . . . . 8 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F)))
3	2	biimpd 153	. . . . . . 7 |- (F e. Fil -> (F e. Fil -> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F)))
4	3	imp 350	. . . . . 6 |- ((F e. Fil /\ F e. Fil) -> ((-. (/) e. F /\ U.F e. F) /\ A.xA.y((x e. F /\ y (_ U.F /\ x (_ y) -> y e. F) /\ A.x e. F A.y e. F (x i^i y) e. F))
5	4	3simp3d 795	. . . . 5 |- ((F e. Fil /\ F e. Fil) -> A.x e. F A.y e. F (x i^i y) e. F)
6	5	ex 373	. . . 4 |- (F e. Fil -> (F e. Fil -> A.x e. F A.y e. F (x i^i y) e. F))
7		ineq1 2206	. . . . . . 7 |- (x = A -> (x i^i y) = (A i^i y))
8		eleq1 1531	. . . . . . 7 |- ((x i^i y) = (A i^i y) -> ((x i^i y) e. F <-> (A i^i y) e. F))
9	7, 8	syl 10	. . . . . 6 |- (x = A -> ((x i^i y) e. F <-> (A i^i y) e. F))
10		ineq2 2207	. . . . . . 7 |- (y = B -> (A i^i y) = (A i^i B))
11		eleq1 1531	. . . . . . 7 |- ((A i^i y) = (A i^i B) -> ((A i^i y) e. F <-> (A i^i B) e. F))
12	10, 11	syl 10	. . . . . 6 |- (y = B -> ((A i^i y) e. F <-> (A i^i B) e. F))
13	9, 12	rcla42v 1876	. . . . 5 |- ((A e. F /\ B e. F) -> (A.x e. F A.y e. F (x i^i y) e. F -> (A i^i B) e. F))
14	13	com12 11	. . . 4 |- (A.x e. F A.y e. F (x i^i y) e. F -> ((A e. F /\ B e. F) -> (A i^i B) e. F))
15	6, 14	syl6 22	. . 3 |- (F e. Fil -> (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) e. F)))
16	15	pm2.43i 64	. 2 |- (F e. Fil -> ((A e. F /\ B e. F) -> (A i^i B) e. F))
17	16	3impib 830	1 |- ((F e. Fil /\ A e. F /\ B e. F) -> (A i^i B) e. F)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774  A.wal 952   = wceq 954   e. wcel 956  A.wral 1642   i^i cin 2042   (_ wss 2043  (/)c0 2276  U.cuni 2498  Filcfil 10467

This theorem is referenced by:  fipfil 10474  filintf 10479

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-in 2047  df-ss 2049  df-uni 2499  df-fil 10468

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