Theorem filint2 10505

Description: A filter is closed under taking finite intersections.

Assertion

Ref	Expression
filint2	|- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> |^|A e. F))

Distinct variable groups:   x,A   x,F

Proof of Theorem filint2

Step	Hyp	Ref	Expression
1		elpw2g 2723	. . . . . . 7 |- (F e. Fil -> (A e. P~F <-> A (_ F))
2		sseq1 2079	. . . . . . . . 9 |- (y = A -> (y (_ F <-> A (_ F))
3		neeq1 1588	. . . . . . . . . . 11 |- (y = A -> (y =/= (/) <-> A =/= (/)))
4		breq1 2618	. . . . . . . . . . . . 13 |- (y = A -> (y ~~ x <-> A ~~ x))
5	4	rexbidv 1662	. . . . . . . . . . . 12 |- (y = A -> (E.x e. om y ~~ x <-> E.x e. om A ~~ x))
6		inteq 2532	. . . . . . . . . . . . 13 |- (y = A -> |^|y = |^|A)
7	6	eleq1d 1538	. . . . . . . . . . . 12 |- (y = A -> (|^|y e. F <-> |^|A e. F))
8	5, 7	imbi12d 625	. . . . . . . . . . 11 |- (y = A -> ((E.x e. om y ~~ x -> |^|y e. F) <-> (E.x e. om A ~~ x -> |^|A e. F)))
9	3, 8	imbi12d 625	. . . . . . . . . 10 |- (y = A -> ((y =/= (/) -> (E.x e. om y ~~ x -> |^|y e. F)) <-> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F))))
10	9	imbi2d 611	. . . . . . . . 9 |- (y = A -> ((F e. Fil -> (y =/= (/) -> (E.x e. om y ~~ x -> |^|y e. F))) <-> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F)))))
11	2, 10	imbi12d 625	. . . . . . . 8 |- (y = A -> ((y (_ F -> (F e. Fil -> (y =/= (/) -> (E.x e. om y ~~ x -> |^|y e. F)))) <-> (A (_ F -> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F))))))
12		eqid 1474	. . . . . . . . . . . . 13 |- U.F = U.F
13	12	isfil 10492	. . . . . . . . . . . 12 |- (F e. Fil -> (F e. Fil <-> ((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F)))
14		3simp3 789	. . . . . . . . . . . . . 14 |- (((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F) -> A.y e. F A.x e. F (y i^i x) e. F)
15		fiint 4543	. . . . . . . . . . . . . 14 |- (A.y e. F A.x e. F (y i^i x) e. F <-> A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
16	14, 15	sylib 198	. . . . . . . . . . . . 13 |- (((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F) -> A.y((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
17	16	19.21bi 1059	. . . . . . . . . . . 12 |- (((-. (/) e. F /\ U.F e. F) /\ A.yA.x((y e. F /\ x (_ U.F /\ y (_ x) -> x e. F) /\ A.y e. F A.x e. F (y i^i x) e. F) -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
18	13, 17	syl6bi 214	. . . . . . . . . . 11 |- (F e. Fil -> (F e. Fil -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F)))
19	18	pm2.43i 64	. . . . . . . . . 10 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ E.x e. om y ~~ x) -> |^|y e. F))
20	19	3expd 849	. . . . . . . . 9 |- (F e. Fil -> (y (_ F -> (y =/= (/) -> (E.x e. om y ~~ x -> |^|y e. F))))
21	20	com12 11	. . . . . . . 8 |- (y (_ F -> (F e. Fil -> (y =/= (/) -> (E.x e. om y ~~ x -> |^|y e. F))))
22	11, 21	vtoclg 1844	. . . . . . 7 |- (A e. P~F -> (A (_ F -> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F)))))
23	1, 22	syl6bir 215	. . . . . 6 |- (F e. Fil -> (A (_ F -> (A (_ F -> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F))))))
24	23	com4l 39	. . . . 5 |- (A (_ F -> (A (_ F -> (F e. Fil -> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F))))))
25	24	pm2.43i 64	. . . 4 |- (A (_ F -> (F e. Fil -> (F e. Fil -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F)))))
26	25	com13 33	. . 3 |- (F e. Fil -> (F e. Fil -> (A (_ F -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F)))))
27	26	pm2.43i 64	. 2 |- (F e. Fil -> (A (_ F -> (A =/= (/) -> (E.x e. om A ~~ x -> |^|A e. F))))
28	27	3impd 846	1 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ E.x e. om A ~~ x) -> |^|A e. F))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957   =/= wne 1583  A.wral 1643  E.wrex 1644   i^i cin 2043   (_ wss 2044  (/)c0 2277  P~cpw 2398  U.cuni 2499  |^|cint 2529   class class class wbr 2615  omcom 3127   ~~ cen 4357  Filcfil 10490

This theorem is referenced by:  cnfilca 10510

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1o 4126  df-oadd 4128  df-er 4254  df-en 4360  df-fil 10491

Copyright terms: Public domain