Theorem neiptopreu 19500

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 19477, innei 19492, elnei 19478 and neissex 19494, 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 19498	. . . 4 |- ( ph -> J e. Top )
9		eqid 2441	. . . . 5 |- U. J = U. J
10	9	toptopon 19301	. . . 4 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
11	8, 10	sylib 196	. . 3 |- ( ph -> J e. (TopOn ` U. J ) )
12	1, 2, 3, 4, 5, 6, 7	neiptopuni 19497	. . . 4 |- ( ph -> X = U. J )
13	12	fveq2d 5856	. . 3 |- ( ph -> (TopOn ` X ) = (TopOn ` U. J ) )
14	11, 13	eleqtrrd 2532	. 2 |- ( ph -> J e. (TopOn ` X ) )
15	1, 2, 3, 4, 5, 6, 7	neiptopnei 19499	. 2 |- ( ph -> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
16		nfv 1692	. . . . . . . . . 10 |- F/ p ( ph /\ j e. (TopOn ` X ) )
17		nfmpt1 4522	. . . . . . . . . . 11 |- F/_ p ( p e. X |-> ( ( nei ` j ) ` { p } ) )
18	17	nfeq2 2620	. . . . . . . . . 10 |- F/ p N = ( p e. X |-> ( ( nei ` j ) ` { p } ) )
19	16, 18	nfan 1912	. . . . . . . . 9 |- F/ p ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
20		nfv 1692	. . . . . . . . 9 |- F/ p b C_ X
21	19, 20	nfan 1912	. . . . . . . 8 |- F/ p ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X )
22		simpllr 758	. . . . . . . . . . 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 461	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) -> b C_ X )
24	23	sselda 3486	. . . . . . . . . . 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		fvex 5862	. . . . . . . . . . . . 13 |- ( ( nei ` j ) ` { p } ) e. _V
27	26	a1i 11	. . . . . . . . . . . 12 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( ( nei ` j ) ` { p } ) e. _V )
28	25, 27	fvmpt2d 5946	. . . . . . . . . . 11 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
29	22, 24, 28	syl2anc 661	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
30	29	eqcomd 2449	. . . . . . . . 9 |- ( ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) /\ p e. b ) -> ( ( nei ` j ) ` { p } ) = ( N ` p ) )
31	30	eleq2d 2511	. . . . . . . 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 ) ) )
32	21, 31	ralbida 2874	. . . . . . 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 ) ) )
33	32	pm5.32da 641	. . . . . 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 ) ) ) )
34		simpllr 758	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> j e. (TopOn ` X ) )
35		simpr 461	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b e. j )
36		toponss 19297	. . . . . . . . 9 |- ( ( j e. (TopOn ` X ) /\ b e. j ) -> b C_ X )
37	34, 35, 36	syl2anc 661	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b C_ X )
38		topontop 19294	. . . . . . . . . . 11 |- ( j e. (TopOn ` X ) -> j e. Top )
39	38	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j e. Top )
40		opnnei 19487	. . . . . . . . . 10 |- ( j e. Top -> ( b e. j <-> A. p e. b b e. ( ( nei ` j ) ` { p } ) ) )
41	39, 40	syl 16	. . . . . . . . 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 } ) ) )
42	41	biimpa 484	. . . . . . . 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 } ) )
43	37, 42	jca 532	. . . . . . 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 } ) ) )
44	41	biimpar 485	. . . . . . . 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 )
45	44	adantrl 715	. . . . . . 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 )
46	43, 45	impbida 830	. . . . . 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 } ) ) ) )
47	1	neipeltop 19496	. . . . . . 7 |- ( b e. J <-> ( b C_ X /\ A. p e. b b e. ( N ` p ) ) )
48	47	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 ) ) ) )
49	33, 46, 48	3bitr4d 285	. . . . 5 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. j <-> b e. J ) )
50	49	eqrdv 2438	. . . 4 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j = J )
51	50	ex 434	. . 3 |- ( ( ph /\ j e. (TopOn ` X ) ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
52	51	ralrimiva 2855	. 2 |- ( ph -> A. j e. (TopOn ` X ) ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
53		simpl 457	. . . . . . 7 |- ( ( j = J /\ p e. X ) -> j = J )
54	53	fveq2d 5856	. . . . . 6 |- ( ( j = J /\ p e. X ) -> ( nei ` j ) = ( nei ` J ) )
55	54	fveq1d 5854	. . . . 5 |- ( ( j = J /\ p e. X ) -> ( ( nei ` j ) ` { p } ) = ( ( nei ` J ) ` { p } ) )
56	55	mpteq2dva 4519	. . . 4 |- ( j = J -> ( p e. X |-> ( ( nei ` j ) ` { p } ) ) = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
57	56	eqeq2d 2455	. . 3 |- ( j = J -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) ) )
58	57	eqreu 3275	. 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 } ) ) )
59	14, 15, 52, 58	syl3anc 1227	1 |- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 972   = wceq 1381   e. wcel 1802   A.wral 2791   E.wrex 2792   E!wreu 2793   {crab 2795   _Vcvv 3093   C_ wss 3458   ~Pcpw 3993   {csn 4010   U.cuni 4230   |-> cmpt 4491   -->wf 5570   ` cfv 5574   ficfi 7868   Topctop 19261  TopOnctopon 19262   neicnei 19464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-fin 7518  df-fi 7869  df-top 19266  df-topon 19269  df-ntr 19387  df-nei 19465

This theorem is referenced by:  ustuqtop  20615