Theorem tailfb 32372

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 32370	. . . 4 |- ( D e. DirRel -> ( tail ` D ) : X --> ~P X )
3		frn 6053	. . . 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 481	. 2 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) C_ ~P X )
6		n0 3931	. . . . 5 |- ( X =/= (/) <-> E. x x e. X )
7		ffn 6045	. . . . . . . 8 |- ( ( tail ` D ) : X --> ~P X -> ( tail ` D ) Fn X )
8		fnfvelrn 6356	. . . . . . . . 9 |- ( ( ( tail ` D ) Fn X /\ x e. X ) -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) )
9	8	ex 450	. . . . . . . 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 3921	. . . . . . 7 |- ( ( ( tail ` D ) ` x ) e. ran ( tail ` D ) -> ran ( tail ` D ) =/= (/) )
12	10, 11	syl6 35	. . . . . 6 |- ( D e. DirRel -> ( x e. X -> ran ( tail ` D ) =/= (/) ) )
13	12	exlimdv 1861	. . . . 5 |- ( D e. DirRel -> ( E. x x e. X -> ran ( tail ` D ) =/= (/) ) )
14	6, 13	syl5bi 232	. . . 4 |- ( D e. DirRel -> ( X =/= (/) -> ran ( tail ` D ) =/= (/) ) )
15	14	imp 445	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) =/= (/) )
16	1	tailini 32371	. . . . . . . 8 |- ( ( D e. DirRel /\ x e. X ) -> x e. ( ( tail ` D ) ` x ) )
17		n0i 3920	. . . . . . . 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 3001	. . . . . 6 |- ( D e. DirRel -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
20	19	adantr 481	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
21		fvelrnb 6243	. . . . . . 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 481	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
24	20, 23	mtbird 315	. . . 4 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. (/) e. ran ( tail ` D ) )
25		df-nel 2898	. . . 4 |- ( (/) e/ ran ( tail ` D ) <-> -. (/) e. ran ( tail ` D ) )
26	24, 25	sylibr 224	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> (/) e/ ran ( tail ` D ) )
27		fvelrnb 6243	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( x e. ran ( tail ` D ) <-> E. u e. X ( ( tail ` D ) ` u ) = x ) )
28		fvelrnb 6243	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( y e. ran ( tail ` D ) <-> E. v e. X ( ( tail ` D ) ` v ) = y ) )
29	27, 28	anbi12d 747	. . . . . . 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 3107	. . . . . . 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 17237	. . . . . . . . . . 11 |- ( ( D e. DirRel /\ u e. X /\ v e. X ) -> E. w e. X ( u D w /\ v D w ) )
33	32	3expb 1266	. . . . . . . . . 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 6356	. . . . . . . . . . . . 13 |- ( ( ( tail ` D ) Fn X /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
36	34, 35	sylan 488	. . . . . . . . . . . 12 |- ( ( D e. DirRel /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
37	36	ad2ant2r 783	. . . . . . . . . . 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 3203	. . . . . . . . . . . . . . . . . . . . . 22 |- x e. _V
39		dirtr 17236	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( u D w /\ w D x ) ) -> u D x )
40	39	exp32 631	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( u D w -> ( w D x -> u D x ) ) )
41	38, 40	mpan2 707	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( u D w -> ( w D x -> u D x ) ) )
42	41	com23 86	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( u D w -> u D x ) ) )
43	42	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( u D w -> u D x ) )
44	43	ad2ant2rl 785	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ w D x ) ) -> ( u D w -> u D x ) )
45		dirtr 17236	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( v D w /\ w D x ) ) -> v D x )
46	45	exp32 631	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( v D w -> ( w D x -> v D x ) ) )
47	38, 46	mpan2 707	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( v D w -> ( w D x -> v D x ) ) )
48	47	com23 86	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( v D w -> v D x ) ) )
49	48	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( v D w -> v D x ) )
50	49	ad2ant2rl 785	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . 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 649	. . . . . . . . . . . . . 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 32369	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ w e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
56	38, 55	mp3an3 1413	. . . . . . . . . . . . . . 15 |- ( ( D e. DirRel /\ w e. X ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
57	56	ad2ant2r 783	. . . . . . . . . . . . . 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 32369	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ u e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
59	38, 58	mp3an3 1413	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ u e. X ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
60	59	adantrr 753	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
61	1	eltail 32369	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ v e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
62	38, 61	mp3an3 1413	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ v e. X ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
63	62	adantrl 752	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
64	60, 63	anbi12d 747	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 3796	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 3609	. . . . . . . . . . 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 3626	. . . . . . . . . . . 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 3309	. . . . . . . . . . 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 693	. . . . . . . . . 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 3035	. . . . . . . . 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 3807	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` u ) = x -> ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) = ( x i^i ( ( tail ` D ) ` v ) ) )
75	74	sseq2d 3633	. . . . . . . . . . 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 3052	. . . . . . . . . 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 3808	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` v ) = y -> ( x i^i ( ( tail ` D ) ` v ) ) = ( x i^i y ) )
78	77	sseq2d 3633	. . . . . . . . . . 11 |- ( ( ( tail ` D ) ` v ) = y -> ( z C_ ( x i^i ( ( tail ` D ) ` v ) ) <-> z C_ ( x i^i y ) ) )
79	78	rexbidv 3052	. . . . . . . . . 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 736	. . . . . . . . 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 235	. . . . . . . 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 3038	. . . . . . 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 233	. . . . . 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 230	. . . . 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 481	. . . 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 2970	. . 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 1242	. 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 7097	. . . . 5 |- ( D e. DirRel -> dom D e. _V )
89	1, 88	syl5eqel 2705	. . . 4 |- ( D e. DirRel -> X e. _V )
90	89	adantr 481	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> X e. _V )
91		isfbas2 21639	. . 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 957	1 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) e. ( fBas ` X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888   DirRelcdir 17228   tailctail 17229   fBascfbas 19734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-dir 17230  df-tail 17231  df-fbas 19743

This theorem is referenced by:  filnetlem4  32376