Theorem filint2 10574

Description: A filter is closed under taking finite intersections.

Assertion

Ref	Expression
filint2	|- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> |^|A e. F))

Proof of Theorem filint2

Step	Hyp	Ref	Expression
1		elpw2g 2727	. . . . . . 7 |- (F e. Fil -> (A e. P~F <-> A (_ F))
2		sseq1 2082	. . . . . . . . 9 |- (y = A -> (y (_ F <-> A (_ F))
3		neeq1 1590	. . . . . . . . . . 11 |- (y = A -> (y =/= (/) <-> A =/= (/)))
4		eleq1 1534	. . . . . . . . . . . 12 |- (y = A -> (y e. Fin <-> A e. Fin))
5		inteq 2536	. . . . . . . . . . . . 13 |- (y = A -> |^|y = |^|A)
6	5	eleq1d 1540	. . . . . . . . . . . 12 |- (y = A -> (|^|y e. F <-> |^|A e. F))
7	4, 6	imbi12d 626	. . . . . . . . . . 11 |- (y = A -> ((y e. Fin -> |^|y e. F) <-> (A e. Fin -> |^|A e. F)))
8	3, 7	imbi12d 626	. . . . . . . . . 10 |- (y = A -> ((y =/= (/) -> (y e. Fin -> |^|y e. F)) <-> (A =/= (/) -> (A e. Fin -> |^|A e. F))))
9	8	imbi2d 612	. . . . . . . . 9 |- (y = A -> ((F e. Fil -> (y =/= (/) -> (y e. Fin -> |^|y e. F))) <-> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F)))))
10	2, 9	imbi12d 626	. . . . . . . 8 |- (y = A -> ((y (_ F -> (F e. Fil -> (y =/= (/) -> (y e. Fin -> |^|y e. F)))) <-> (A (_ F -> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F))))))
11		eqid 1475	. . . . . . . . . . . . 13 |- U.F = U.F
12	11	isfil 10558	. . . . . . . . . . . 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)))
13		3simp3 790	. . . . . . . . . . . . . 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)
14		fiint 4559	. . . . . . . . . . . . . 14 |- (A.y e. F A.x e. F (y i^i x) e. F <-> A.y((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
15	13, 14	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 =/= (/) /\ y e. Fin) -> |^|y e. F))
16	15	19.21bi 1060	. . . . . . . . . . . 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 =/= (/) /\ y e. Fin) -> |^|y e. F))
17	12, 16	syl6bi 214	. . . . . . . . . . 11 |- (F e. Fil -> (F e. Fil -> ((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F)))
18	17	pm2.43i 64	. . . . . . . . . 10 |- (F e. Fil -> ((y (_ F /\ y =/= (/) /\ y e. Fin) -> |^|y e. F))
19	18	3expd 850	. . . . . . . . 9 |- (F e. Fil -> (y (_ F -> (y =/= (/) -> (y e. Fin -> |^|y e. F))))
20	19	com12 11	. . . . . . . 8 |- (y (_ F -> (F e. Fil -> (y =/= (/) -> (y e. Fin -> |^|y e. F))))
21	10, 20	vtoclg 1847	. . . . . . 7 |- (A e. P~F -> (A (_ F -> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F)))))
22	1, 21	syl6bir 215	. . . . . 6 |- (F e. Fil -> (A (_ F -> (A (_ F -> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F))))))
23	22	com4l 39	. . . . 5 |- (A (_ F -> (A (_ F -> (F e. Fil -> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F))))))
24	23	pm2.43i 64	. . . 4 |- (A (_ F -> (F e. Fil -> (F e. Fil -> (A =/= (/) -> (A e. Fin -> |^|A e. F)))))
25	24	com13 33	. . 3 |- (F e. Fil -> (F e. Fil -> (A (_ F -> (A =/= (/) -> (A e. Fin -> |^|A e. F)))))
26	25	pm2.43i 64	. 2 |- (F e. Fil -> (A (_ F -> (A =/= (/) -> (A e. Fin -> |^|A e. F))))
27	26	3impd 847	1 |- (F e. Fil -> ((A (_ F /\ A =/= (/) /\ A e. Fin) -> |^|A e. F))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645   i^i cin 2046   (_ wss 2047  (/)c0 2280  P~cpw 2401  U.cuni 2503  |^|cint 2533  Fincfn 4367  Filcfil 10556

This theorem is referenced by:  cnfilca 10583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-er 4261  df-en 4368  df-fin 4371  df-fil 10557

Copyright terms: Public domain