Theorem ustuqtop 22050

Description: For a given uniform structure U on a set X, there is a unique topology j such that the set ran ( v e. U |-> ( v " { p } ) ) is the filter of the neighborhoods of p for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	|- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )

Assertion

Ref	Expression
ustuqtop	|- ( U e. (UnifOn ` X ) -> E! j e. (TopOn ` X ) A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) )

Distinct variable groups:   v, p, U   X, p, v, j   j, N, p   v, j, U   j, X

Allowed substitution hint:   N( v)

Proof of Theorem ustuqtop

Dummy variables a b c r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( p = r -> ( N ` p ) = ( N ` r ) )
2	1	eleq2d 2687	. . . . . 6 |- ( p = r -> ( c e. ( N ` p ) <-> c e. ( N ` r ) ) )
3	2	cbvralv 3171	. . . . 5 |- ( A. p e. c c e. ( N ` p ) <-> A. r e. c c e. ( N ` r ) )
4		eleq1 2689	. . . . . 6 |- ( c = a -> ( c e. ( N ` p ) <-> a e. ( N ` p ) ) )
5	4	raleqbi1dv 3146	. . . . 5 |- ( c = a -> ( A. p e. c c e. ( N ` p ) <-> A. p e. a a e. ( N ` p ) ) )
6	3, 5	syl5bbr 274	. . . 4 |- ( c = a -> ( A. r e. c c e. ( N ` r ) <-> A. p e. a a e. ( N ` p ) ) )
7	6	cbvrabv 3199	. . 3 |- { c e. ~P X | A. r e. c c e. ( N ` r ) } = { a e. ~P X | A. p e. a a e. ( N ` p ) }
8		utopustuq.1	. . . 4 |- N = ( p e. X |-> ran ( v e. U |-> ( v " { p } ) ) )
9	8	ustuqtop0 22044	. . 3 |- ( U e. (UnifOn ` X ) -> N : X --> ~P ~P X )
10	8	ustuqtop1 22045	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )
11	8	ustuqtop2 22046	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
12	8	ustuqtop3 22047	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )
13	8	ustuqtop4 22048	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ p e. X ) /\ a e. ( N ` p ) ) -> E. b e. ( N ` p ) A. x e. b a e. ( N ` x ) )
14	8	ustuqtop5 22049	. . 3 |- ( ( U e. (UnifOn ` X ) /\ p e. X ) -> X e. ( N ` p ) )
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 20937	. 2 |- ( U e. (UnifOn ` X ) -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
16	9	feqmptd 6249	. . . . 5 |- ( U e. (UnifOn ` X ) -> N = ( p e. X |-> ( N ` p ) ) )
17	16	eqeq1d 2624	. . . 4 |- ( U e. (UnifOn ` X ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> ( p e. X |-> ( N ` p ) ) = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) )
18		fvex 6201	. . . . . 6 |- ( N ` p ) e. _V
19	18	rgenw 2924	. . . . 5 |- A. p e. X ( N ` p ) e. _V
20		mpteqb 6299	. . . . 5 |- ( A. p e. X ( N ` p ) e. _V -> ( ( p e. X |-> ( N ` p ) ) = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) ) )
21	19, 20	ax-mp 5	. . . 4 |- ( ( p e. X |-> ( N ` p ) ) = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) )
22	17, 21	syl6bb 276	. . 3 |- ( U e. (UnifOn ` X ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) ) )
23	22	reubidv 3126	. 2 |- ( U e. (UnifOn ` X ) -> ( E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> E! j e. (TopOn ` X ) A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) ) )
24	15, 23	mpbid 222	1 |- ( U e. (UnifOn ` X ) -> E! j e. (TopOn ` X ) A. p e. X ( N ` p ) = ( ( nei ` j ) ` { p } ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   {crab 2916   _Vcvv 3200   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ran crn 5115   "cima 5117   ` cfv 5888  TopOnctopon 20715   neicnei 20901  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-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-3or 1038  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-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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-ust 22004

This theorem is referenced by: (None)