Theorem neiptopreu 19927

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 19904, innei 19919, elnei 19905 and neissex 19921, 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 19925	. . . 4 |- ( ph -> J e. Top )
9		eqid 2402	. . . . 5 |- U. J = U. J
10	9	toptopon 19726	. . . 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 19924	. . . 4 |- ( ph -> X = U. J )
13	12	fveq2d 5853	. . 3 |- ( ph -> (TopOn ` X ) = (TopOn ` U. J ) )
14	11, 13	eleqtrrd 2493	. 2 |- ( ph -> J e. (TopOn ` X ) )
15	1, 2, 3, 4, 5, 6, 7	neiptopnei 19926	. 2 |- ( ph -> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
16		nfv 1728	. . . . . . . . . 10 |- F/ p ( ph /\ j e. (TopOn ` X ) )
17		nfmpt1 4484	. . . . . . . . . . 11 |- F/_ p ( p e. X |-> ( ( nei ` j ) ` { p } ) )
18	17	nfeq2 2581	. . . . . . . . . 10 |- F/ p N = ( p e. X |-> ( ( nei ` j ) ` { p } ) )
19	16, 18	nfan 1956	. . . . . . . . 9 |- F/ p ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
20		nfv 1728	. . . . . . . . 9 |- F/ p b C_ X
21	19, 20	nfan 1956	. . . . . . . 8 |- F/ p ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X )
22		simpllr 761	. . . . . . . . . . 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 459	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) -> b C_ X )
24	23	sselda 3442	. . . . . . . . . . 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 5859	. . . . . . . . . . . . 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 5943	. . . . . . . . . . 11 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
29	22, 24, 28	syl2anc 659	. . . . . . . . . 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 2410	. . . . . . . . 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 2472	. . . . . . . 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 2837	. . . . . . 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 639	. . . . . 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 761	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> j e. (TopOn ` X ) )
35		simpr 459	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b e. j )
36		toponss 19722	. . . . . . . . 9 |- ( ( j e. (TopOn ` X ) /\ b e. j ) -> b C_ X )
37	34, 35, 36	syl2anc 659	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b C_ X )
38		topontop 19719	. . . . . . . . . . 11 |- ( j e. (TopOn ` X ) -> j e. Top )
39	38	ad2antlr 725	. . . . . . . . . 10 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j e. Top )
40		opnnei 19914	. . . . . . . . . 10 |- ( j e. Top -> ( b e. j <-> A. p e. b b e. ( ( nei ` j ) ` { p } ) ) )
41	39, 40	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 } ) ) )
42	41	biimpa 482	. . . . . . . 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 530	. . . . . . 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 483	. . . . . . . 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 714	. . . . . . 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 833	. . . . . 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 19923	. . . . . . 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 2399	. . . 4 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j = J )
51	50	ex 432	. . 3 |- ( ( ph /\ j e. (TopOn ` X ) ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
52	51	ralrimiva 2818	. 2 |- ( ph -> A. j e. (TopOn ` X ) ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
53		simpl 455	. . . . . . 7 |- ( ( j = J /\ p e. X ) -> j = J )
54	53	fveq2d 5853	. . . . . 6 |- ( ( j = J /\ p e. X ) -> ( nei ` j ) = ( nei ` J ) )
55	54	fveq1d 5851	. . . . 5 |- ( ( j = J /\ p e. X ) -> ( ( nei ` j ) ` { p } ) = ( ( nei ` J ) ` { p } ) )
56	55	mpteq2dva 4481	. . . 4 |- ( j = J -> ( p e. X |-> ( ( nei ` j ) ` { p } ) ) = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
57	56	eqeq2d 2416	. . 3 |- ( j = J -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) ) )
58	57	eqreu 3241	. 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 1230	1 |- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2754   E.wrex 2755   E!wreu 2756   {crab 2758   _Vcvv 3059   C_ wss 3414   ~Pcpw 3955   {csn 3972   U.cuni 4191   |-> cmpt 4453   -->wf 5565   ` cfv 5569   ficfi 7904   Topctop 19686  TopOnctopon 19687   neicnei 19891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-fin 7558  df-fi 7905  df-top 19691  df-topon 19694  df-ntr 19813  df-nei 19892

This theorem is referenced by:  ustuqtop  21041