Theorem tailfb 26296

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 26294	. . . 4 |- ( D e. DirRel -> ( tail ` D ) : X --> ~P X )
3		frn 5556	. . . 4 |- ( ( tail ` D ) : X --> ~P X -> ran ( tail ` D ) C_ ~P X )
4	2, 3	syl 16	. . 3 |- ( D e. DirRel -> ran ( tail ` D ) C_ ~P X )
5	4	adantr 452	. 2 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) C_ ~P X )
6		n0 3597	. . . . 5 |- ( X =/= (/) <-> E. x x e. X )
7		ffn 5550	. . . . . . . 8 |- ( ( tail ` D ) : X --> ~P X -> ( tail ` D ) Fn X )
8		fnfvelrn 5826	. . . . . . . . 9 |- ( ( ( tail ` D ) Fn X /\ x e. X ) -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) )
9	8	ex 424	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( x e. X -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) ) )
10	2, 7, 9	3syl 19	. . . . . . 7 |- ( D e. DirRel -> ( x e. X -> ( ( tail ` D ) ` x ) e. ran ( tail ` D ) ) )
11		ne0i 3594	. . . . . . 7 |- ( ( ( tail ` D ) ` x ) e. ran ( tail ` D ) -> ran ( tail ` D ) =/= (/) )
12	10, 11	syl6 31	. . . . . 6 |- ( D e. DirRel -> ( x e. X -> ran ( tail ` D ) =/= (/) ) )
13	12	exlimdv 1643	. . . . 5 |- ( D e. DirRel -> ( E. x x e. X -> ran ( tail ` D ) =/= (/) ) )
14	6, 13	syl5bi 209	. . . 4 |- ( D e. DirRel -> ( X =/= (/) -> ran ( tail ` D ) =/= (/) ) )
15	14	imp 419	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) =/= (/) )
16	1	tailini 26295	. . . . . . . 8 |- ( ( D e. DirRel /\ x e. X ) -> x e. ( ( tail ` D ) ` x ) )
17		n0i 3593	. . . . . . . 8 |- ( x e. ( ( tail ` D ) ` x ) -> -. ( ( tail ` D ) ` x ) = (/) )
18	16, 17	syl 16	. . . . . . 7 |- ( ( D e. DirRel /\ x e. X ) -> -. ( ( tail ` D ) ` x ) = (/) )
19	18	nrexdv 2769	. . . . . 6 |- ( D e. DirRel -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
20	19	adantr 452	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. E. x e. X ( ( tail ` D ) ` x ) = (/) )
21		fvelrnb 5733	. . . . . . 7 |- ( ( tail ` D ) Fn X -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
22	2, 7, 21	3syl 19	. . . . . 6 |- ( D e. DirRel -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
23	22	adantr 452	. . . . 5 |- ( ( D e. DirRel /\ X =/= (/) ) -> ( (/) e. ran ( tail ` D ) <-> E. x e. X ( ( tail ` D ) ` x ) = (/) ) )
24	20, 23	mtbird 293	. . . 4 |- ( ( D e. DirRel /\ X =/= (/) ) -> -. (/) e. ran ( tail ` D ) )
25		df-nel 2570	. . . 4 |- ( (/) e/ ran ( tail ` D ) <-> -. (/) e. ran ( tail ` D ) )
26	24, 25	sylibr 204	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> (/) e/ ran ( tail ` D ) )
27		fvelrnb 5733	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( x e. ran ( tail ` D ) <-> E. u e. X ( ( tail ` D ) ` u ) = x ) )
28		fvelrnb 5733	. . . . . . . 8 |- ( ( tail ` D ) Fn X -> ( y e. ran ( tail ` D ) <-> E. v e. X ( ( tail ` D ) ` v ) = y ) )
29	27, 28	anbi12d 692	. . . . . . 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 19	. . . . . 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 2835	. . . . . . 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 14637	. . . . . . . . . . 11 |- ( ( D e. DirRel /\ u e. X /\ v e. X ) -> E. w e. X ( u D w /\ v D w ) )
33	32	3expb 1154	. . . . . . . . . 10 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> E. w e. X ( u D w /\ v D w ) )
34	2, 7	syl 16	. . . . . . . . . . . . 13 |- ( D e. DirRel -> ( tail ` D ) Fn X )
35		fnfvelrn 5826	. . . . . . . . . . . . 13 |- ( ( ( tail ` D ) Fn X /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
36	34, 35	sylan 458	. . . . . . . . . . . 12 |- ( ( D e. DirRel /\ w e. X ) -> ( ( tail ` D ) ` w ) e. ran ( tail ` D ) )
37	36	ad2ant2r 728	. . . . . . . . . . 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 2919	. . . . . . . . . . . . . . . . . . . . . 22 |- x e. _V
39		dirtr 14636	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( u D w /\ w D x ) ) -> u D x )
40	39	exp32 589	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( u D w -> ( w D x -> u D x ) ) )
41	38, 40	mpan2 653	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( u D w -> ( w D x -> u D x ) ) )
42	41	com23 74	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( u D w -> u D x ) ) )
43	42	imp 419	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( u D w -> u D x ) )
44	43	ad2ant2rl 730	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) /\ ( w e. X /\ w D x ) ) -> ( u D w -> u D x ) )
45		dirtr 14636	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( D e. DirRel /\ x e. _V ) /\ ( v D w /\ w D x ) ) -> v D x )
46	45	exp32 589	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( D e. DirRel /\ x e. _V ) -> ( v D w -> ( w D x -> v D x ) ) )
47	38, 46	mpan2 653	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. DirRel -> ( v D w -> ( w D x -> v D x ) ) )
48	47	com23 74	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. DirRel -> ( w D x -> ( v D w -> v D x ) ) )
49	48	imp 419	. . . . . . . . . . . . . . . . . . 19 |- ( ( D e. DirRel /\ w D x ) -> ( v D w -> v D x ) )
50	49	ad2ant2rl 730	. . . . . . . . . . . . . . . . . 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 547	. . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . 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 74	. . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . 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 26293	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ w e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
56	38, 55	mp3an3 1268	. . . . . . . . . . . . . . 15 |- ( ( D e. DirRel /\ w e. X ) -> ( x e. ( ( tail ` D ) ` w ) <-> w D x ) )
57	56	ad2ant2r 728	. . . . . . . . . . . . . 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 26293	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ u e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
59	38, 58	mp3an3 1268	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ u e. X ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
60	59	adantrr 698	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` u ) <-> u D x ) )
61	1	eltail 26293	. . . . . . . . . . . . . . . . . 18 |- ( ( D e. DirRel /\ v e. X /\ x e. _V ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
62	38, 61	mp3an3 1268	. . . . . . . . . . . . . . . . 17 |- ( ( D e. DirRel /\ v e. X ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
63	62	adantrl 697	. . . . . . . . . . . . . . . 16 |- ( ( D e. DirRel /\ ( u e. X /\ v e. X ) ) -> ( x e. ( ( tail ` D ) ` v ) <-> v D x ) )
64	60, 63	anbi12d 692	. . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 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 3490	. . . . . . . . . . . . 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 219	. . . . . . . . . . . 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 3314	. . . . . . . . . . 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 3329	. . . . . . . . . . . 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 3012	. . . . . . . . . . 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 643	. . . . . . . . . 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 2794	. . . . . . . . 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 3495	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` u ) = x -> ( ( ( tail ` D ) ` u ) i^i ( ( tail ` D ) ` v ) ) = ( x i^i ( ( tail ` D ) ` v ) ) )
75	74	sseq2d 3336	. . . . . . . . . . 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 2687	. . . . . . . . . 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 3496	. . . . . . . . . . . 12 |- ( ( ( tail ` D ) ` v ) = y -> ( x i^i ( ( tail ` D ) ` v ) ) = ( x i^i y ) )
78	77	sseq2d 3336	. . . . . . . . . . 11 |- ( ( ( tail ` D ) ` v ) = y -> ( z C_ ( x i^i ( ( tail ` D ) ` v ) ) <-> z C_ ( x i^i y ) ) )
79	78	rexbidv 2687	. . . . . . . . . 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 681	. . . . . . . . 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 212	. . . . . . . 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 2797	. . . . . . 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 210	. . . . . 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 207	. . . . 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 452	. . . 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 2757	. . 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 1134	. 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 5089	. . . . 5 |- ( D e. DirRel -> dom D e. _V )
89	1, 88	syl5eqel 2488	. . . 4 |- ( D e. DirRel -> X e. _V )
90	89	adantr 452	. . 3 |- ( ( D e. DirRel /\ X =/= (/) ) -> X e. _V )
91		isfbas2 17820	. . 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 16	. 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 889	1 |- ( ( D e. DirRel /\ X =/= (/) ) -> ran ( tail ` D ) e. ( fBas ` X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   e/ wnel 2568   A.wral 2666   E.wrex 2667   _Vcvv 2916   i^i cin 3279   C_ wss 3280   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172   dom cdm 4837   ran crn 4838   Fn wfn 5408   -->wf 5409   ` cfv 5413   DirRelcdir 14628   tailctail 14629   fBascfbas 16644

This theorem is referenced by:  filnetlem4  26300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-dir 14630  df-tail 14631  df-fbas 16654