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Theorem neiptopreu 18737
Description: If, to each element  P of a set  X, we associate a set  ( N `  P ) fulfilling the properties Vi, Vii, Viii and property Viv 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
StepHypRef 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
) )
81, 2, 3, 4, 5, 6, 7neiptoptop 18735 . . . 4  |-  ( ph  ->  J  e.  Top )
9 eqid 2443 . . . . 5  |-  U. J  =  U. J
109toptopon 18538 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
118, 10sylib 196 . . 3  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
121, 2, 3, 4, 5, 6, 7neiptopuni 18734 . . . 4  |-  ( ph  ->  X  =  U. J
)
1312fveq2d 5695 . . 3  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. J ) )
1411, 13eleqtrrd 2520 . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
151, 2, 3, 4, 5, 6, 7neiptopnei 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 } ) )
1817nfeq2 2590 . . . . . . . . . 10  |-  F/ p  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )
1916, 18nfan 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
2119, 20nfan 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 )
2423sselda 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
2726a1i 11 . . . . . . . . . . . 12  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  (
( nei `  j
) `  { p } )  e.  _V )
2825, 27fvmpt2d 5783 . . . . . . . . . . 11  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
2922, 24, 28syl2anc 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 }
) )
3029eqcomd 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 ) )
3130eleq2d 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 ) ) )
3221, 31ralbida 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 ) ) )
3332pm5.32da 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 )
3734, 35, 36syl2anc 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 )
3938ad2antlr 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 } ) ) )
4139, 40syl 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 }
) ) )
4241biimpa 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 } ) )
4337, 42jca 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 } ) ) )
4441biimpar 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 )
4544adantrl 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 )
4643, 45impbida 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 } ) ) ) )
471neipeltop 18733 . . . . . . 7  |-  ( b  e.  J  <->  ( b  C_  X  /\  A. p  e.  b  b  e.  ( N `  p ) ) )
4847a1i 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 ) ) ) )
4933, 46, 483bitr4d 285 . . . . 5  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  b  e.  J
) )
5049eqrdv 2441 . . . 4  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  =  J )
5150ex 434 . . 3  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  -> 
j  =  J ) )
5251ralrimiva 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 )
5453fveq2d 5695 . . . . . 6  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( nei `  j
)  =  ( nei `  J ) )
5554fveq1d 5693 . . . . 5  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( ( nei `  j
) `  { p } )  =  ( ( nei `  J
) `  { p } ) )
5655mpteq2dva 4378 . . . 4  |-  ( j  =  J  ->  (
p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) )
5756eqeq2d 2454 . . 3  |-  ( j  =  J  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  N  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) ) )
5857eqreu 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 }
) ) )
5914, 15, 52, 58syl3anc 1218 1  |-  ( ph  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )
Colors of variables: wff setvar class
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
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