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Theorem neiptopreu 20226
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 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
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 20224 . . . 4  |-  ( ph  ->  J  e.  Top )
9 eqid 2471 . . . . 5  |-  U. J  =  U. J
109toptopon 20025 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
118, 10sylib 201 . . 3  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
121, 2, 3, 4, 5, 6, 7neiptopuni 20223 . . . 4  |-  ( ph  ->  X  =  U. J
)
1312fveq2d 5883 . . 3  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. J ) )
1411, 13eleqtrrd 2552 . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
151, 2, 3, 4, 5, 6, 7neiptopnei 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 } ) )
1817nfeq2 2627 . . . . . . . . . 10  |-  F/ p  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )
1916, 18nfan 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
2119, 20nfan 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 )
2423sselda 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
2726a1i 11 . . . . . . . . . . . 12  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  (
( nei `  j
) `  { p } )  e.  _V )
2825, 27fvmpt2d 5974 . . . . . . . . . . 11  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
2922, 24, 28syl2anc 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 }
) )
3029eqcomd 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 ) )
3130eleq2d 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 ) ) )
3221, 31ralbida 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 ) ) )
3332pm5.32da 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 )
3734, 35, 36syl2anc 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 )
3938ad2antlr 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 } ) ) )
4139, 40syl 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 }
) ) )
4241biimpa 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 } ) )
4337, 42jca 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 } ) ) )
4441biimpar 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 )
4544adantrl 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 )
4643, 45impbida 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 } ) ) ) )
471neipeltop 20222 . . . . . . 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 293 . . . . 5  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  b  e.  J
) )
5049eqrdv 2469 . . . 4  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  =  J )
5150ex 441 . . 3  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  -> 
j  =  J ) )
5251ralrimiva 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 )
5453fveq2d 5883 . . . . . 6  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( nei `  j
)  =  ( nei `  J ) )
5554fveq1d 5881 . . . . 5  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( ( nei `  j
) `  { p } )  =  ( ( nei `  J
) `  { p } ) )
5655mpteq2dva 4482 . . . 4  |-  ( j  =  J  ->  (
p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) )
5756eqeq2d 2481 . . 3  |-  ( j  =  J  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  N  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) ) )
5857eqreu 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 }
) ) )
5914, 15, 52, 58syl3anc 1292 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 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
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