Theorem neiptopreu 18737

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 18714, innei 18729, elnei 18715 and neissex 18731, 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 18735	. . . 4 |- ( ph -> J e. Top )
9		eqid 2443	. . . . 5 |- U. J = U. J
10	9	toptopon 18538	. . . 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 18734	. . . 4 |- ( ph -> X = U. J )
13	12	fveq2d 5695	. . 3 |- ( ph -> (TopOn ` X ) = (TopOn ` U. J ) )
14	11, 13	eleqtrrd 2520	. 2 |- ( ph -> J e. (TopOn ` X ) )
15	1, 2, 3, 4, 5, 6, 7	neiptopnei 18736	. 2 |- ( ph -> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
16		nfv 1673	. . . . . . . . . 10 |- F/ p ( ph /\ j e. (TopOn ` X ) )
17		nfmpt1 4381	. . . . . . . . . . 11 |- F/_ p ( p e. X |-> ( ( nei ` j ) ` { p } ) )
18	17	nfeq2 2590	. . . . . . . . . 10 |- F/ p N = ( p e. X |-> ( ( nei ` j ) ` { p } ) )
19	16, 18	nfan 1861	. . . . . . . . 9 |- F/ p ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
20		nfv 1673	. . . . . . . . 9 |- F/ p b C_ X
21	19, 20	nfan 1861	. . . . . . . 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 3356	. . . . . . . . . . 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 5701	. . . . . . . . . . . . 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 5783	. . . . . . . . . . 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 2448	. . . . . . . . 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 2510	. . . . . . . 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 2729	. . . . . . 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 18534	. . . . . . . . 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 18531	. . . . . . . . . . 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 18724	. . . . . . . . . 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 828	. . . . . 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 18733	. . . . . . 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 2441	. . . 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 2799	. 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 5695	. . . . . 6 |- ( ( j = J /\ p e. X ) -> ( nei ` j ) = ( nei ` J ) )
55	54	fveq1d 5693	. . . . 5 |- ( ( j = J /\ p e. X ) -> ( ( nei ` j ) ` { p } ) = ( ( nei ` J ) ` { p } ) )
56	55	mpteq2dva 4378	. . . 4 |- ( j = J -> ( p e. X |-> ( ( nei ` j ) ` { p } ) ) = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
57	56	eqeq2d 2454	. . 3 |- ( j = J -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) ) )
58	57	eqreu 3151	. 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 1218	1 |- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2715   E.wrex 2716   E!wreu 2717   {crab 2719   _Vcvv 2972   C_ wss 3328   ~Pcpw 3860   {csn 3877   U.cuni 4091   e. cmpt 4350   -->wf 5414   ` cfv 5418   ficfi 7660   Topctop 18498  TopOnctopon 18499   neicnei 18701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-fin 7314  df-fi 7661  df-top 18503  df-topon 18506  df-ntr 18624  df-nei 18702

This theorem is referenced by:  ustuqtop  19821