Theorem tailfb 31104

Description: The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

Hypothesis

Ref	Expression
tailfb.1	|- X = dom D

Assertion

Ref	Expression
tailfb	|- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) e. ( fBas ` X ) )

Proof of Theorem tailfb

Dummy variables v u w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tailfb.1	. . . . 5 |- X = dom D
2	1	tailf 31102	. . . 4 |- ( D e. DirRel -> ( tail ` D ) : X --> ~P X )
3		frn 5747	. . . 4 |- ( ( tail ` D ) : X --> ~P X -> ran ( tail ` D ) C_ ~P X )
4	2, 3	syl 17	. . 3 |- ( D e. DirRel -> ran ( tail ` D ) C_ ~P X )
5	4	adantr 472	. 2 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) C_ ~P X )
6		n0 3732	. . . . 5 |- ( X =/= (/) <-> E. x x e. X )
7		ffn 5739	. . . . . . . 8 |- ( ( tail ` D ) : X --> ~P X -> ( tail ` D ) Fn X )
8		fnfvelrn 6034	. . . . . . . . 9 |- ( ( ( tail ` D ) Fn X /\ x e. X ) -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) )
9	8	ex 441	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( x e. X -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) ) )
10	2, 7, 9	3syl 18	. . . . . . 7 |- ( D e. DirRel -> ( x e. X -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) ) )
11		ne0i 3728	. . . . . . 7 |- ( ( ( tail ` D ) ` x ) e. ran ( tail ` D ) -> ran ( tail ` D ) =/= (/) )
12	10, 11	syl6 33	. . . . . 6 |- ( D e. DirRel -> ( x e. X -> ran ( tail ` D ) =/= (/) ) )
13	12	exlimdv 1787	. . . . 5 |- ( D e. DirRel -> ( E. x x e. X -> ran ( tail ` D ) =/= (/) ) )
14	6, 13	syl5bi 225	. . . 4 |- ( D e. DirRel -> ( X =/= (/) -> ran ( tail ` D ) =/= (/) ) )
15	14	imp 436	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) =/= (/) )
16	1	tailini 31103	. . . . . . . 8 |- ( ( D e. DirRel /\ x e. X ) -> x e. ( ( tail ` D ) ` x ) )
17		n0i 3727	. . . . . . . 8 |- ( x e. ( ( tail ` D ) ` x ) -> -. ( ( tail ` D ) ` x ) = (/) )
18	16, 17	syl 17	. . . . . . 7 |- ( ( D e. DirRel /\ x e. X ) -> -. ( ( tail ` D ) ` x ) = (/) )
19	18	nrexdv 2842	. . . . . 6 |- ( D e. DirRel -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
20	19	adantr 472	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
21		fvelrnb 5926	. . . . . . 7 |- ( ( tail ` D ) Fn X -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
22	2, 7, 21	3syl 18	. . . . . 6 |- ( D e. DirRel -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
23	22	adantr 472	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
24	20, 23	mtbird 308	. . . 4 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. (/) e. ran ( tail ` D ) )
25		df-nel 2644	. . . 4 |- ( (/) e/ ran ( tail ` D ) <-> -. (/) e. ran ( tail ` D ) )
26	24, 25	sylibr 217	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> (/) e/ ran ( tail ` D ) )
27		fvelrnb 5926	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( x e. ran ( tail ` D ) <-> E. u e. X ( ( tail ` D ) ` u ) = x ) )
28		fvelrnb 5926	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( y e. ran ( tail ` D ) <-> E. v e. X ( ( tail ` D ) ` v ) = y ) )
29	27, 28	anbi12d 725	. . . . . . 7 |- ( ( tail ` D ) Fn X -> ( ( x e. ran ( tail ` D ) /\ y e. ran ( tail ` D ) ) <-> ( E. u e. X ( ( tail ` D ) ` u ) = x /\ E. v e. X ( ( tail ` D ) ` v ) = y ) ) )
30	2, 7, 29	3syl 18	. . . . . 6 |- ( D e. DirRel -> ( ( x e. ran ( tail ` D ) /\ y e. ran ( tail ` D ) ) <-> ( E. u e. X ( ( tail ` D ) ` u ) = x /\ E. v e. X ( ( tail ` D ) ` v ) = y ) ) )
31		reeanv 2944	. . . . . . 7 |- ( E. u e. X E. v e. X ( ( ( tail ` D ) ` u ) = x /\ ( ( tail ` D ) ` v ) = y ) <-> ( E. u e. X ( ( tail ` D ) ` u ) = x /\ E. v e. X ( ( tail ` D ) ` v ) = y ) )
32	1	dirge 16561	. . . . . . . . . . 11 |- ( ( D e. DirRel /\ u e. X /\ v e. X ) -> E. w e. X ( u D w /\ v D w ) )
33	32	3expb 1232	. . . . . . . . . 10 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> E. w e. X ( u D w /\ v D w ) )
34	2, 7	syl 17	. . . . . . . . . . . . 13 |- ( D e. DirRel -> ( tail ` D ) Fn X )
35		fnfvelrn 6034	. . . . . . . . . . . . 13 |- ( ( ( tail ` D ) Fn X /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
36	34, 35	sylan 479	. . . . . . . . . . . 12 |- ( ( D e. DirRel /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
37	36	ad2ant2r 761	. . . . . . . . . . 11 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
38		vex 3034	. . . . . . . . . . . . . . . . . . . . . 22 |- x e. _V
39		dirtr 16560	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( u D w /\ w D x ) ) -> u D x )
40	39	exp32 616	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( u D w -> ( w D x -> u D x ) ) )
41	38, 40	mpan2 685	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( u D w -> ( w D x -> u D x ) ) )
42	41	com23 80	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( u D w -> u D x ) ) )
43	42	imp 436	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( u D w -> u D x ) )
44	43	ad2ant2rl 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ w D x ) ) -> ( u D w -> u D x ) )
45		dirtr 16560	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( v D w /\ w D x ) ) -> v D x )
46	45	exp32 616	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( v D w -> ( w D x -> v D x ) ) )
47	38, 46	mpan2 685	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( v D w -> ( w D x -> v D x ) ) )
48	47	com23 80	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( v D w -> v D x ) ) )
49	48	imp 436	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( v D w -> v D x ) )
50	49	ad2ant2rl 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ w D x ) ) -> ( v D w -> v D x ) )
51	44, 50	anim12d 572	. . . . . . . . . . . . . . . . 17 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ w D x ) ) -> ( ( u D w /\ v D w ) -> ( u D x /\ v D x ) ) )
52	51	expr 626	. . . . . . . . . . . . . . . 16 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ w e. X ) -> ( w D x -> ( ( u D w /\ v D w ) -> ( u D x /\ v D x ) ) ) )
53	52	com23 80	. . . . . . . . . . . . . . 15 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ w e. X ) -> ( ( u D w /\ v D w ) -> ( w D x -> ( u D x /\ v D x ) ) ) )
54	53	impr 631	. . . . . . . . . . . . . 14 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( w D x -> ( u D x /\ v D x ) ) )
55	1	eltail 31101	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ w e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
56	38, 55	mp3an3 1379	. . . . . . . . . . . . . . 15 |- ( ( D e. DirRel /\ w e. X ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
57	56	ad2ant2r 761	. . . . . . . . . . . . . 14 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
58	1	eltail 31101	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ u e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
59	38, 58	mp3an3 1379	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ u e. X ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
60	59	adantrr 731	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
61	1	eltail 31101	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ v e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
62	38, 61	mp3an3 1379	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ v e. X ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
63	62	adantrl 730	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
64	60, 63	anbi12d 725	. . . . . . . . . . . . . . 15 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( ( x e. ( ( tail ` D ) ` u ) /\ x e. ( ( tail ` D ) ` v ) ) <-> ( u D x /\ v D x ) ) )
65	64	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( ( x e. ( ( tail ` D ) ` u ) /\ x e. ( ( tail ` D ) ` v ) ) <-> ( u D x /\ v D x ) ) )
66	54, 57, 65	3imtr4d 276	. . . . . . . . . . . . 13 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( x e. ( ( tail ` D ) ` w ) -> ( x e. ( ( tail ` D ) ` u ) /\ x e. ( ( tail ` D ) ` v ) ) ) )
67		elin 3608	. . . . . . . . . . . . 13 |- ( x e. ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) <-> ( x e. ( ( tail ` D ) ` u ) /\ x e. ( ( tail ` D ) ` v ) ) )
68	66, 67	syl6ibr 235	. . . . . . . . . . . 12 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( x e. ( ( tail ` D ) ` w ) -> x e. ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) ) )
69	68	ssrdv 3424	. . . . . . . . . . 11 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> ( ( tail ` D ) ` w ) C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) )
70		sseq1 3439	. . . . . . . . . . . 12 |- ( z = ( ( tail ` D ) ` w ) -> ( z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) <-> ( ( tail ` D ) ` w ) C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) ) )
71	70	rspcev 3136	. . . . . . . . . . 11 |- ( ( ( ( tail ` D ) ` w ) e. ran ( tail ` D ) /\ ( ( tail ` D ) ` w ) C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) ) -> E. z e. ran ( tail ` D ) z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) )
72	37, 69, 71	syl2anc 673	. . . . . . . . . 10 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ ( u D w /\ v D w ) ) ) -> E. z e. ran ( tail ` D ) z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) )
73	33, 72	rexlimddv 2875	. . . . . . . . 9 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> E. z e. ran ( tail ` D ) z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) )
74		ineq1 3618	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` u ) = x -> ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) = ( x i^i ( ( tail ` D ) ` v ) ) )
75	74	sseq2d 3446	. . . . . . . . . . 11 |- ( ( ( tail ` D ) ` u ) = x -> ( z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) <-> z C_ ( x i^i ( ( tail ` D ) ` v ) ) ) )
76	75	rexbidv 2892	. . . . . . . . . 10 |- ( ( ( tail ` D ) ` u ) = x -> ( E. z e. ran ( tail ` D ) z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) <-> E. z e. ran ( tail ` D ) z C_ ( x i^i ( ( tail ` D ) ` v ) ) ) )
77		ineq2 3619	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` v ) = y -> ( x i^i ( ( tail ` D ) ` v ) ) = ( x i^i y ) )
78	77	sseq2d 3446	. . . . . . . . . . 11 |- ( ( ( tail ` D ) ` v ) = y -> ( z C_ ( x i^i ( ( tail ` D ) ` v ) ) <-> z C_ ( x i^i y ) ) )
79	78	rexbidv 2892	. . . . . . . . . 10 |- ( ( ( tail ` D ) ` v ) = y -> ( E. z e. ran ( tail ` D ) z C_ ( x i^i ( ( tail ` D ) ` v ) ) <-> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
80	76, 79	sylan9bb 714	. . . . . . . . 9 |- ( ( ( ( tail ` D ) ` u ) = x /\ ( ( tail ` D ) ` v ) = y ) -> ( E. z e. ran ( tail ` D ) z C_ ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) <-> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
81	73, 80	syl5ibcom 228	. . . . . . . 8 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( ( ( ( tail ` D ) ` u ) = x /\ ( ( tail ` D ) ` v ) = y ) -> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
82	81	rexlimdvva 2878	. . . . . . 7 |- ( D e. DirRel -> ( E. u e. X E. v e. X ( ( ( tail ` D ) ` u ) = x /\ ( ( tail ` D ) ` v ) = y ) -> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
83	31, 82	syl5bir 226	. . . . . 6 |- ( D e. DirRel -> ( ( E. u e. X ( ( tail ` D ) ` u ) = x /\ E. v e. X ( ( tail ` D ) ` v ) = y ) -> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
84	30, 83	sylbid 223	. . . . 5 |- ( D e. DirRel -> ( ( x e. ran ( tail ` D ) /\ y e. ran ( tail ` D ) ) -> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
85	84	adantr 472	. . . 4 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( ( x e. ran ( tail ` D ) /\ y e. ran ( tail ` D ) ) -> E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
86	85	ralrimivv 2813	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> A. x e. ran ( tail ` D ) A. y e. ran ( tail ` D ) E. z e. ran ( tail ` D ) z C_ ( x i^i y ) )
87	15, 26, 86	3jca 1210	. 2 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( ran ( tail ` D ) =/= (/) /\ (/) e/ ran ( tail ` D ) /\ A. x e. ran ( tail ` D ) A. y e. ran ( tail ` D ) E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) )
88		dmexg 6743	. . . . 5 |- ( D e. DirRel -> dom D e. _V )
89	1, 88	syl5eqel 2553	. . . 4 |- ( D e. DirRel -> X e. _V )
90	89	adantr 472	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> X e. _V )
91		isfbas2 20928	. . 3 |- ( X e. _V -> ( ran ( tail ` D ) e. ( fBas ` X ) <-> ( ran ( tail ` D ) C_ ~P X /\ ( ran ( tail ` D ) =/= (/) /\ (/) e/ ran ( tail ` D ) /\ A. x e. ran ( tail ` D ) A. y e. ran ( tail ` D ) E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) ) ) )
92	90, 91	syl 17	. 2 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( ran ( tail ` D ) e. ( fBas ` X ) <-> ( ran ( tail ` D ) C_ ~P X /\ ( ran ( tail ` D ) =/= (/) /\ (/) e/ ran ( tail ` D ) /\ A. x e. ran ( tail ` D ) A. y e. ran ( tail ` D ) E. z e. ran ( tail ` D ) z C_ ( x i^i y ) ) ) ) )
93	5, 87, 92	mpbir2and 936	1 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) e. ( fBas ` X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   e/ wnel 2642   A.wral 2756   E.wrex 2757   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   class class class wbr 4395   dom cdm 4839   ran crn 4840   Fn wfn 5584   -->wf 5585   ` cfv 5589   DirRelcdir 16552   tailctail 16553   fBascfbas 19035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-dir 16554  df-tail 16555  df-fbas 19044

This theorem is referenced by:  filnetlem4  31108