Theorem opnfbas 21646

Description: The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Hypothesis

Ref	Expression
opnfbas.1	|- X = U. J

Assertion

Ref	Expression
opnfbas	|- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )

Distinct variable groups:   x, J   x, S   x, X

Proof of Theorem opnfbas

Dummy variables s r t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . 4 |- { x e. J | S C_ x } C_ J
2		opnfbas.1	. . . . . 6 |- X = U. J
3	2	eqimss2i 3660	. . . . 5 |- U. J C_ X
4		sspwuni 4611	. . . . 5 |- ( J C_ ~P X <-> U. J C_ X )
5	3, 4	mpbir 221	. . . 4 |- J C_ ~P X
6	1, 5	sstri 3612	. . 3 |- { x e. J | S C_ x } C_ ~P X
7	6	a1i 11	. 2 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } C_ ~P X )
8	2	topopn 20711	. . . . . . 7 |- ( J e. Top -> X e. J )
9	8	anim1i 592	. . . . . 6 |- ( ( J e. Top /\ S C_ X ) -> ( X e. J /\ S C_ X ) )
10	9	3adant3 1081	. . . . 5 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X e. J /\ S C_ X ) )
11		sseq2 3627	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
12	11	elrab 3363	. . . . 5 |- ( X e. { x e. J | S C_ x } <-> ( X e. J /\ S C_ X ) )
13	10, 12	sylibr 224	. . . 4 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> X e. { x e. J | S C_ x } )
14		ne0i 3921	. . . 4 |- ( X e. { x e. J | S C_ x } -> { x e. J | S C_ x } =/= (/) )
15	13, 14	syl 17	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } =/= (/) )
16		ss0 3974	. . . . . . 7 |- ( S C_ (/) -> S = (/) )
17	16	necon3ai 2819	. . . . . 6 |- ( S =/= (/) -> -. S C_ (/) )
18	17	3ad2ant3 1084	. . . . 5 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> -. S C_ (/) )
19	18	intnand 962	. . . 4 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> -. ( (/) e. J /\ S C_ (/) ) )
20		df-nel 2898	. . . . 5 |- ( (/) e/ { x e. J | S C_ x } <-> -. (/) e. { x e. J | S C_ x } )
21		sseq2 3627	. . . . . . 7 |- ( x = (/) -> ( S C_ x <-> S C_ (/) ) )
22	21	elrab 3363	. . . . . 6 |- ( (/) e. { x e. J | S C_ x } <-> ( (/) e. J /\ S C_ (/) ) )
23	22	notbii 310	. . . . 5 |- ( -. (/) e. { x e. J | S C_ x } <-> -. ( (/) e. J /\ S C_ (/) ) )
24	20, 23	bitr2i 265	. . . 4 |- ( -. ( (/) e. J /\ S C_ (/) ) <-> (/) e/ { x e. J | S C_ x } )
25	19, 24	sylib 208	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> (/) e/ { x e. J | S C_ x } )
26		sseq2 3627	. . . . . . 7 |- ( x = r -> ( S C_ x <-> S C_ r ) )
27	26	elrab 3363	. . . . . 6 |- ( r e. { x e. J | S C_ x } <-> ( r e. J /\ S C_ r ) )
28		sseq2 3627	. . . . . . 7 |- ( x = s -> ( S C_ x <-> S C_ s ) )
29	28	elrab 3363	. . . . . 6 |- ( s e. { x e. J | S C_ x } <-> ( s e. J /\ S C_ s ) )
30	27, 29	anbi12i 733	. . . . 5 |- ( ( r e. { x e. J | S C_ x } /\ s e. { x e. J | S C_ x } ) <-> ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) )
31		simpl 473	. . . . . . . . . . 11 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> J e. Top )
32		simprll 802	. . . . . . . . . . 11 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> r e. J )
33		simprrl 804	. . . . . . . . . . 11 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> s e. J )
34		inopn 20704	. . . . . . . . . . 11 |- ( ( J e. Top /\ r e. J /\ s e. J ) -> ( r i^i s ) e. J )
35	31, 32, 33, 34	syl3anc 1326	. . . . . . . . . 10 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> ( r i^i s ) e. J )
36		ssin 3835	. . . . . . . . . . . . 13 |- ( ( S C_ r /\ S C_ s ) <-> S C_ ( r i^i s ) )
37	36	biimpi 206	. . . . . . . . . . . 12 |- ( ( S C_ r /\ S C_ s ) -> S C_ ( r i^i s ) )
38	37	ad2ant2l 782	. . . . . . . . . . 11 |- ( ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) -> S C_ ( r i^i s ) )
39	38	adantl 482	. . . . . . . . . 10 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> S C_ ( r i^i s ) )
40	35, 39	jca 554	. . . . . . . . 9 |- ( ( J e. Top /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> ( ( r i^i s ) e. J /\ S C_ ( r i^i s ) ) )
41	40	3ad2antl1 1223	. . . . . . . 8 |- ( ( ( J e. Top /\ S C_ X /\ S =/= (/) ) /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> ( ( r i^i s ) e. J /\ S C_ ( r i^i s ) ) )
42		sseq2 3627	. . . . . . . . 9 |- ( x = ( r i^i s ) -> ( S C_ x <-> S C_ ( r i^i s ) ) )
43	42	elrab 3363	. . . . . . . 8 |- ( ( r i^i s ) e. { x e. J | S C_ x } <-> ( ( r i^i s ) e. J /\ S C_ ( r i^i s ) ) )
44	41, 43	sylibr 224	. . . . . . 7 |- ( ( ( J e. Top /\ S C_ X /\ S =/= (/) ) /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> ( r i^i s ) e. { x e. J | S C_ x } )
45		ssid 3624	. . . . . . 7 |- ( r i^i s ) C_ ( r i^i s )
46		sseq1 3626	. . . . . . . 8 |- ( t = ( r i^i s ) -> ( t C_ ( r i^i s ) <-> ( r i^i s ) C_ ( r i^i s ) ) )
47	46	rspcev 3309	. . . . . . 7 |- ( ( ( r i^i s ) e. { x e. J | S C_ x } /\ ( r i^i s ) C_ ( r i^i s ) ) -> E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) )
48	44, 45, 47	sylancl 694	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ S =/= (/) ) /\ ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) ) -> E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) )
49	48	ex 450	. . . . 5 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( ( r e. J /\ S C_ r ) /\ ( s e. J /\ S C_ s ) ) -> E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) )
50	30, 49	syl5bi 232	. . . 4 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( r e. { x e. J | S C_ x } /\ s e. { x e. J | S C_ x } ) -> E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) )
51	50	ralrimivv 2970	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> A. r e. { x e. J | S C_ x } A. s e. { x e. J | S C_ x } E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) )
52	15, 25, 51	3jca 1242	. 2 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( { x e. J | S C_ x } =/= (/) /\ (/) e/ { x e. J | S C_ x } /\ A. r e. { x e. J | S C_ x } A. s e. { x e. J | S C_ x } E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) )
53		isfbas2 21639	. . . 4 |- ( X e. J -> ( { x e. J | S C_ x } e. ( fBas ` X ) <-> ( { x e. J | S C_ x } C_ ~P X /\ ( { x e. J | S C_ x } =/= (/) /\ (/) e/ { x e. J | S C_ x } /\ A. r e. { x e. J | S C_ x } A. s e. { x e. J | S C_ x } E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) ) ) )
54	8, 53	syl 17	. . 3 |- ( J e. Top -> ( { x e. J | S C_ x } e. ( fBas ` X ) <-> ( { x e. J | S C_ x } C_ ~P X /\ ( { x e. J | S C_ x } =/= (/) /\ (/) e/ { x e. J | S C_ x } /\ A. r e. { x e. J | S C_ x } A. s e. { x e. J | S C_ x } E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) ) ) )
55	54	3ad2ant1 1082	. 2 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( { x e. J | S C_ x } e. ( fBas ` X ) <-> ( { x e. J | S C_ x } C_ ~P X /\ ( { x e. J | S C_ x } =/= (/) /\ (/) e/ { x e. J | S C_ x } /\ A. r e. { x e. J | S C_ x } A. s e. { x e. J | S C_ x } E. t e. { x e. J | S C_ x } t C_ ( r i^i s ) ) ) ) )
56	7, 52, 55	mpbir2and 957	1 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   fBascfbas 19734   Topctop 20698

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

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-top 20699

This theorem is referenced by:  neifg  32366