Theorem neiptopreu 20226

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 20203, innei 20218, elnei 20204 and neissex 20220, 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 20224	. . . 4 |- ( ph -> J e. Top )
9		eqid 2471	. . . . 5 |- U. J = U. J
10	9	toptopon 20025	. . . 4 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
11	8, 10	sylib 201	. . 3 |- ( ph -> J e. (TopOn ` U. J ) )
12	1, 2, 3, 4, 5, 6, 7	neiptopuni 20223	. . . 4 |- ( ph -> X = U. J )
13	12	fveq2d 5883	. . 3 |- ( ph -> (TopOn ` X ) = (TopOn ` U. J ) )
14	11, 13	eleqtrrd 2552	. 2 |- ( ph -> J e. (TopOn ` X ) )
15	1, 2, 3, 4, 5, 6, 7	neiptopnei 20225	. 2 |- ( ph -> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
16		nfv 1769	. . . . . . . . . 10 |- F/ p ( ph /\ j e. (TopOn ` X ) )
17		nfmpt1 4485	. . . . . . . . . . 11 |- F/_ p ( p e. X |-> ( ( nei ` j ) ` { p } ) )
18	17	nfeq2 2627	. . . . . . . . . 10 |- F/ p N = ( p e. X |-> ( ( nei ` j ) ` { p } ) )
19	16, 18	nfan 2031	. . . . . . . . 9 |- F/ p ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )
20		nfv 1769	. . . . . . . . 9 |- F/ p b C_ X
21	19, 20	nfan 2031	. . . . . . . 8 |- F/ p ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X )
22		simpllr 777	. . . . . . . . . . 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 468	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b C_ X ) -> b C_ X )
24	23	sselda 3418	. . . . . . . . . . 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 5889	. . . . . . . . . . . . 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 5974	. . . . . . . . . . 11 |- ( ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) /\ p e. X ) -> ( N ` p ) = ( ( nei ` j ) ` { p } ) )
29	22, 24, 28	syl2anc 673	. . . . . . . . . 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 2477	. . . . . . . . 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 2534	. . . . . . . 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 2825	. . . . . . 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 653	. . . . . 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 777	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> j e. (TopOn ` X ) )
35		simpr 468	. . . . . . . . 9 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b e. j )
36		toponss 20021	. . . . . . . . 9 |- ( ( j e. (TopOn ` X ) /\ b e. j ) -> b C_ X )
37	34, 35, 36	syl2anc 673	. . . . . . . 8 |- ( ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) /\ b e. j ) -> b C_ X )
38		topontop 20018	. . . . . . . . . . 11 |- ( j e. (TopOn ` X ) -> j e. Top )
39	38	ad2antlr 741	. . . . . . . . . 10 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j e. Top )
40		opnnei 20213	. . . . . . . . . 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 492	. . . . . . . 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 541	. . . . . . 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 493	. . . . . . . 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 730	. . . . . . 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 850	. . . . . 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 20222	. . . . . . 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 293	. . . . 5 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> ( b e. j <-> b e. J ) )
50	49	eqrdv 2469	. . . 4 |- ( ( ( ph /\ j e. (TopOn ` X ) ) /\ N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) ) -> j = J )
51	50	ex 441	. . 3 |- ( ( ph /\ j e. (TopOn ` X ) ) -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
52	51	ralrimiva 2809	. 2 |- ( ph -> A. j e. (TopOn ` X ) ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) -> j = J ) )
53		simpl 464	. . . . . . 7 |- ( ( j = J /\ p e. X ) -> j = J )
54	53	fveq2d 5883	. . . . . 6 |- ( ( j = J /\ p e. X ) -> ( nei ` j ) = ( nei ` J ) )
55	54	fveq1d 5881	. . . . 5 |- ( ( j = J /\ p e. X ) -> ( ( nei ` j ) ` { p } ) = ( ( nei ` J ) ` { p } ) )
56	55	mpteq2dva 4482	. . . 4 |- ( j = J -> ( p e. X |-> ( ( nei ` j ) ` { p } ) ) = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) )
57	56	eqeq2d 2481	. . 3 |- ( j = J -> ( N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) <-> N = ( p e. X |-> ( ( nei ` J ) ` { p } ) ) ) )
58	57	eqreu 3218	. 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 1292	1 |- ( ph -> E! j e. (TopOn ` X ) N = ( p e. X |-> ( ( nei ` j ) ` { p } ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   E!wreu 2758   {crab 2760   _Vcvv 3031   C_ wss 3390   ~Pcpw 3942   {csn 3959   U.cuni 4190   |-> cmpt 4454   -->wf 5585   ` cfv 5589   ficfi 7942   Topctop 19994  TopOnctopon 19995   neicnei 20190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-top 19998  df-topon 20000  df-ntr 20112  df-nei 20191

This theorem is referenced by:  ustuqtop  21339