Theorem ustfilxp 22016

Description: A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ustfilxp	|- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> U e. ( Fil ` ( X X. X ) ) )

Proof of Theorem ustfilxp

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> X e. _V )
2		isust 22007	. . . . . . 7 |- ( X e. _V -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
3	1, 2	syl 17	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( U e. (UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
4	3	ibi 256	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
5	4	adantl 482	. . . 4 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
6	5	simp1d 1073	. . 3 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> U C_ ~P ( X X. X ) )
7	5	simp2d 1074	. . . . 5 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> ( X X. X ) e. U )
8		ne0i 3921	. . . . 5 |- ( ( X X. X ) e. U -> U =/= (/) )
9	7, 8	syl 17	. . . 4 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> U =/= (/) )
10	5	simp3d 1075	. . . . . . . . . 10 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) )
11	10	r19.21bi 2932	. . . . . . . . 9 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) )
12	11	simp3d 1075	. . . . . . . 8 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) )
13	12	simp1d 1073	. . . . . . 7 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> ( _I |` X ) C_ v )
14		vex 3203	. . . . . . . . . . . . 13 |- w e. _V
15		opelresi 5408	. . . . . . . . . . . . 13 |- ( w e. _V -> ( <. w , w >. e. ( _I |` X ) <-> w e. X ) )
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 |- ( <. w , w >. e. ( _I |` X ) <-> w e. X )
17	16	biimpri 218	. . . . . . . . . . 11 |- ( w e. X -> <. w , w >. e. ( _I |` X ) )
18	17	rgen 2922	. . . . . . . . . 10 |- A. w e. X <. w , w >. e. ( _I |` X )
19		r19.2z 4060	. . . . . . . . . 10 |- ( ( X =/= (/) /\ A. w e. X <. w , w >. e. ( _I |` X ) ) -> E. w e. X <. w , w >. e. ( _I |` X ) )
20	18, 19	mpan2 707	. . . . . . . . 9 |- ( X =/= (/) -> E. w e. X <. w , w >. e. ( _I |` X ) )
21	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> E. w e. X <. w , w >. e. ( _I |` X ) )
22		ne0i 3921	. . . . . . . . 9 |- ( <. w , w >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
23	22	rexlimivw 3029	. . . . . . . 8 |- ( E. w e. X <. w , w >. e. ( _I |` X ) -> ( _I |` X ) =/= (/) )
24	21, 23	syl 17	. . . . . . 7 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> ( _I |` X ) =/= (/) )
25		ssn0 3976	. . . . . . 7 |- ( ( ( _I |` X ) C_ v /\ ( _I |` X ) =/= (/) ) -> v =/= (/) )
26	13, 24, 25	syl2anc 693	. . . . . 6 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> v =/= (/) )
27	26	nelrdva 3417	. . . . 5 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> -. (/) e. U )
28		df-nel 2898	. . . . 5 |- ( (/) e/ U <-> -. (/) e. U )
29	27, 28	sylibr 224	. . . 4 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> (/) e/ U )
30	11	simp2d 1074	. . . . . . . . 9 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> A. w e. U ( v i^i w ) e. U )
31	30	r19.21bi 2932	. . . . . . . 8 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) /\ w e. U ) -> ( v i^i w ) e. U )
32	14	inex2 4800	. . . . . . . . . 10 |- ( v i^i w ) e. _V
33	32	pwid 4174	. . . . . . . . 9 |- ( v i^i w ) e. ~P ( v i^i w )
34	33	a1i 11	. . . . . . . 8 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) /\ w e. U ) -> ( v i^i w ) e. ~P ( v i^i w ) )
35	31, 34	elind 3798	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) /\ w e. U ) -> ( v i^i w ) e. ( U i^i ~P ( v i^i w ) ) )
36		ne0i 3921	. . . . . . 7 |- ( ( v i^i w ) e. ( U i^i ~P ( v i^i w ) ) -> ( U i^i ~P ( v i^i w ) ) =/= (/) )
37	35, 36	syl 17	. . . . . 6 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) /\ w e. U ) -> ( U i^i ~P ( v i^i w ) ) =/= (/) )
38	37	ralrimiva 2966	. . . . 5 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) )
39	38	ralrimiva 2966	. . . 4 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> A. v e. U A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) )
40	9, 29, 39	3jca 1242	. . 3 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> ( U =/= (/) /\ (/) e/ U /\ A. v e. U A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) ) )
41		xpexg 6960	. . . . . 6 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
42	1, 1, 41	syl2anc 693	. . . . 5 |- ( U e. (UnifOn ` X ) -> ( X X. X ) e. _V )
43		isfbas 21633	. . . . 5 |- ( ( X X. X ) e. _V -> ( U e. ( fBas ` ( X X. X ) ) <-> ( U C_ ~P ( X X. X ) /\ ( U =/= (/) /\ (/) e/ U /\ A. v e. U A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) ) ) ) )
44	42, 43	syl 17	. . . 4 |- ( U e. (UnifOn ` X ) -> ( U e. ( fBas ` ( X X. X ) ) <-> ( U C_ ~P ( X X. X ) /\ ( U =/= (/) /\ (/) e/ U /\ A. v e. U A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) ) ) ) )
45	44	adantl 482	. . 3 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> ( U e. ( fBas ` ( X X. X ) ) <-> ( U C_ ~P ( X X. X ) /\ ( U =/= (/) /\ (/) e/ U /\ A. v e. U A. w e. U ( U i^i ~P ( v i^i w ) ) =/= (/) ) ) ) )
46	6, 40, 45	mpbir2and 957	. 2 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> U e. ( fBas ` ( X X. X ) ) )
47		n0 3931	. . . . 5 |- ( ( U i^i ~P w ) =/= (/) <-> E. v v e. ( U i^i ~P w ) )
48		elin 3796	. . . . . . 7 |- ( v e. ( U i^i ~P w ) <-> ( v e. U /\ v e. ~P w ) )
49		selpw 4165	. . . . . . . 8 |- ( v e. ~P w <-> v C_ w )
50	49	anbi2i 730	. . . . . . 7 |- ( ( v e. U /\ v e. ~P w ) <-> ( v e. U /\ v C_ w ) )
51	48, 50	bitri 264	. . . . . 6 |- ( v e. ( U i^i ~P w ) <-> ( v e. U /\ v C_ w ) )
52	51	exbii 1774	. . . . 5 |- ( E. v v e. ( U i^i ~P w ) <-> E. v ( v e. U /\ v C_ w ) )
53	47, 52	bitri 264	. . . 4 |- ( ( U i^i ~P w ) =/= (/) <-> E. v ( v e. U /\ v C_ w ) )
54	11	simp1d 1073	. . . . . . . 8 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) -> A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) )
55	54	r19.21bi 2932	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ v e. U ) /\ w e. ~P ( X X. X ) ) -> ( v C_ w -> w e. U ) )
56	55	an32s 846	. . . . . 6 |- ( ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ w e. ~P ( X X. X ) ) /\ v e. U ) -> ( v C_ w -> w e. U ) )
57	56	expimpd 629	. . . . 5 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ w e. ~P ( X X. X ) ) -> ( ( v e. U /\ v C_ w ) -> w e. U ) )
58	57	exlimdv 1861	. . . 4 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ w e. ~P ( X X. X ) ) -> ( E. v ( v e. U /\ v C_ w ) -> w e. U ) )
59	53, 58	syl5bi 232	. . 3 |- ( ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) /\ w e. ~P ( X X. X ) ) -> ( ( U i^i ~P w ) =/= (/) -> w e. U ) )
60	59	ralrimiva 2966	. 2 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> A. w e. ~P ( X X. X ) ( ( U i^i ~P w ) =/= (/) -> w e. U ) )
61		isfil 21651	. 2 |- ( U e. ( Fil ` ( X X. X ) ) <-> ( U e. ( fBas ` ( X X. X ) ) /\ A. w e. ~P ( X X. X ) ( ( U i^i ~P w ) =/= (/) -> w e. U ) ) )
62	46, 60, 61	sylanbrc 698	1 |- ( ( X =/= (/) /\ U e. (UnifOn ` X ) ) -> U e. ( Fil ` ( X X. X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   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   <.cop 4183   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   fBascfbas 19734   Filcfil 21649  UnifOncust 22003

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-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-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-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-fv 5896  df-fbas 19743  df-fil 21650  df-ust 22004

This theorem is referenced by: (None)