Theorem neiptopreu 20937

Description: If, to each element P of a set X, we associate a set ( N ` P ) fulfilling the properties V<sub>i</sub>, V<sub>ii</sub>, V<sub>iii</sub> and property V<sub>iv</sub> of [BourbakiTop1] p. I.2. , corresponding to ssnei 20914, innei 20929, elnei 20915 and neissex 20931, then there is a unique topology j such that for any point p, ( N ` p ) is the set of neighborhoods of p. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that the additional condition that X is a neighborhood of all points was added. (Contributed by Thierry Arnoux, 6-Jan-2018.)

Hypotheses

Ref	Expression
neiptop.o	|- J = { a e. ~P X | A. p e. a a e. ( N ` p ) }
neiptop.0	|- ( ph -> N : X --> ~P ~P X )
neiptop.1	|- ( ( ( ( ph /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )
neiptop.2	|- ( ( ph /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
neiptop.3	|- ( ( ( ph /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )
neiptop.4	|- ( ( ( ph /\ p e. X ) /\ a e. ( N ` p ) ) -> E. b e. ( N ` p ) A. q e. b a e. ( N ` q ) )
neiptop.5	|- ( ( ph /\ p e. X ) -> X e. ( N ` p ) )

Assertion

Ref	Expression
neiptopreu	|- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Distinct variable groups:   p, a, N   X, a, b, p   J, a, p   X, p   ph, p   N, b   X, b   ph, a, b, q, p   N, p, q   X, q   ph, q   j, a, b, J, p   j, q, N   j, X   ph, j

Allowed substitution hint:   J( q)

Proof of Theorem neiptopreu

Step	Hyp	Ref	Expression
1		neiptop.o	. . . . 5 |- J = { a e. ~P X | A. p e. a a e. ( N ` p ) }
2		neiptop.0	. . . . 5 |- ( ph -> N : X --> ~P ~P X )
3		neiptop.1	. . . . 5 |- ( ( ( ( ph /\ p e. X ) /\ a C_ b /\ b C_ X ) /\ a e. ( N ` p ) ) -> b e. ( N ` p ) )
4		neiptop.2	. . . . 5 |- ( ( ph /\ p e. X ) -> ( fi ` ( N ` p ) ) C_ ( N ` p ) )
5		neiptop.3	. . . . 5 |- ( ( ( ph /\ p e. X ) /\ a e. ( N ` p ) ) -> p e. a )
6		neiptop.4	. . . . 5 |- ( ( ( ph /\ p e. X ) /\ a e. ( N ` p ) ) -> E. b e. ( N ` p ) A. q e. b a e. ( N ` q ) )
7		neiptop.5	. . . . 5 |- ( ( ph /\ p e. X ) -> X e. ( N ` p ) )
8	1, 2, 3, 4, 5, 6, 7	neiptoptop 20935	. . . 4 |- ( ph -> J e. Top )
9		eqid 2622	. . . . 5 |- U. J = U. J
10	9	toptopon 20722	. . . 4 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
11	8, 10	sylib 208	. . 3 |- ( ph -> J e. (TopOn ` U. J ) )
12	1, 2, 3, 4, 5, 6, 7	neiptopuni 20934	. . . 4 |- ( ph -> X = U. J )
13	12	fveq2d 6195	. . 3 |- ( ph -> (TopOn ` X ) = (TopOn ` U. J ) )
14	11, 13	eleqtrrd 2704	. 2 |- ( ph -> J e. (TopOn ` X ) )
15	1, 2, 3, 4, 5, 6, 7	neiptopnei 20936	. 2 |- ( ph -> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
16		nfv 1843	. . . . . . . . . 10 |- F/ p ( ph /\ j e. (TopOn ` X ) )
17		nfmpt1 4747	. . . . . . . . . . 11 |- F/_ p ( p e. X |-> ( ( nei ` j ) ` { p } ) )
18	17	nfeq2 2780	. . . . . . . . . 10 |- F/ p N = ( p e. X |-> ( ( nei ` j ) ` { p } ) )
19	16, 18	nfan 1828	. . . . . . . . 9 |- F/ p ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
20		nfv 1843	. . . . . . . . 9 |- F/ p b C_ X
21	19, 20	nfan 1828	. . . . . . . 8 |- F/ p ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X )
22		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
23		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) -> b C_ X )
24	23	sselda 3603	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> p e. X )
25		id 22	. . . . . . . . . . . 12 |- ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
26		fvexd 6203	. . . . . . . . . . . 12 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( ( nei ` j ) ` { p } ) e. _V )
27	25, 26	fvmpt2d 6293	. . . . . . . . . . 11 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
28	22, 24, 27	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
29	28	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> ( ( nei ` j ) ` { p } ) = ( N ` p ) )
30	29	eleq2d 2687	. . . . . . . 8 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> ( b e. ( ( nei ` j ) ` { p } ) <-> b e. ( N ` p ) ) )
31	21, 30	ralbida 2982	. . . . . . 7 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) -> ( A. p e. b b e. ( ( nei ` j ) ` { p } ) <-> A. p e. b b e. ( N ` p ) ) )
32	31	pm5.32da 673	. . . . . 6 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( ( b C_ X /\ A. p e. b b e. ( ( nei ` j ) ` { p } ) ) <-> ( b C_ X /\ A. p e. b b e. ( N ` p ) ) ) )
33		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> j e. (TopOn ` X ) )
34		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b e. j )
35		toponss 20731	. . . . . . . . 9 |- ( ( j e. (TopOn ` X ) /\ b e. j ) -> b C_ X )
36	33, 34, 35	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b C_ X )
37		topontop 20718	. . . . . . . . . . 11 |- ( j e. (TopOn ` X ) -> j e. Top )
38	37	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j e. Top )
39		opnnei 20924	. . . . . . . . . 10 |- ( j e. Top -> ( b e. j <-> A. p e. b b e. ( ( nei ` j ) ` { p } ) ) )
40	38, 39	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. j <-> A. p e. b b e. ( ( nei ` j ) ` { p } ) ) )
41	40	biimpa 501	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> A. p e. b b e. ( ( nei ` j ) ` { p } ) )
42	36, 41	jca 554	. . . . . . 7 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> ( b C_ X /\ A. p e. b b e. ( ( nei ` j ) ` { p } ) ) )
43	40	biimpar 502	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ A. p e. b b e. ( ( nei ` j ) ` { p } ) ) -> b e. j )
44	43	adantrl 752	. . . . . . 7 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ ( b C_ X /\ A. p e. b b e. ( ( nei ` j ) ` { p } ) ) ) -> b e. j )
45	42, 44	impbida 877	. . . . . 6 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. j <-> ( b C_ X /\ A. p e. b b e. ( ( nei ` j ) ` { p } ) ) ) )
46	1	neipeltop 20933	. . . . . . 7 |- ( b e. J <-> ( b C_ X /\ A. p e. b b e. ( N ` p ) ) )
47	46	a1i 11	. . . . . 6 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. J <-> ( b C_ X /\ A. p e. b b e. ( N ` p ) ) ) )
48	32, 45, 47	3bitr4d 300	. . . . 5 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. j <-> b e. J ) )
49	48	eqrdv 2620	. . . 4 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j = J )
50	49	ex 450	. . 3 |- ( ( ph /\ j e. (TopOn ` X ) ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
51	50	ralrimiva 2966	. 2 |- ( ph -> A. j e. (TopOn ` X ) ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
52		simpl 473	. . . . . . 7 |- ( ( j = J /\ p e. X ) -> j = J )
53	52	fveq2d 6195	. . . . . 6 |- ( ( j = J /\ p e. X ) -> ( nei ` j ) = ( nei ` J ) )
54	53	fveq1d 6193	. . . . 5 |- ( ( j = J /\ p e. X ) -> ( ( nei ` j ) ` { p } ) = ( ( nei ` J ) ` { p } ) )
55	54	mpteq2dva 4744	. . . 4 |- ( j = J -> ( p e. X |-> ( ( nei ` j ) ` { p } ) ) = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
56	55	eqeq2d 2632	. . 3 |- ( j = J -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) ) )
57	56	eqreu 3398	. 2 |- ( ( J e. (TopOn ` X ) /\ N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) /\ A. j e. (TopOn ` X ) ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) ) -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
58	14, 15, 51, 57	syl3anc 1326	1 |- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   -->wf 5884   ` cfv 5888   ficfi 8316   Topctop 20698  TopOnctopon 20715   neicnei 20901

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

This theorem is referenced by:  ustuqtop  22050