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Theorem neibastop3 31018
Description: The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
neibastop1.1  |-  ( ph  ->  X  e.  V )
neibastop1.2  |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/)
} ) )
neibastop1.3  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
)  /\  w  e.  ( F `  x ) ) )  ->  (
( F `  x
)  i^i  ~P (
v  i^i  w )
)  =/=  (/) )
neibastop1.4  |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `  x )  i^i  ~P o )  =/=  (/) }
neibastop1.5  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
) ) )  ->  x  e.  v )
neibastop1.6  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t 
( ( F `  y )  i^i  ~P v )  =/=  (/) )
Assertion
Ref Expression
neibastop3  |-  ( ph  ->  E! j  e.  (TopOn `  X ) A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )
Distinct variable groups:    t, n, v, y, j, x    j, J    x, n, J, v, y    t, o, v, w, x, y, j, F, n    ph, j, n, o, t, v, w, x, y    j, X, n, o, t, v, w, x, y
Allowed substitution hints:    J( w, t, o)    V( x, y, w, v, t, j, n, o)

Proof of Theorem neibastop3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 neibastop1.1 . . . 4  |-  ( ph  ->  X  e.  V )
2 neibastop1.2 . . . 4  |-  ( ph  ->  F : X --> ( ~P ~P X  \  { (/)
} ) )
3 neibastop1.3 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
)  /\  w  e.  ( F `  x ) ) )  ->  (
( F `  x
)  i^i  ~P (
v  i^i  w )
)  =/=  (/) )
4 neibastop1.4 . . . 4  |-  J  =  { o  e.  ~P X  |  A. x  e.  o  ( ( F `  x )  i^i  ~P o )  =/=  (/) }
51, 2, 3, 4neibastop1 31015 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
6 neibastop1.5 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
) ) )  ->  x  e.  v )
7 neibastop1.6 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  X  /\  v  e.  ( F `  x
) ) )  ->  E. t  e.  ( F `  x ) A. y  e.  t 
( ( F `  y )  i^i  ~P v )  =/=  (/) )
81, 2, 3, 4, 6, 7neibastop2 31017 . . . . . . . 8  |-  ( (
ph  /\  z  e.  X )  ->  (
n  e.  ( ( nei `  J ) `
 { z } )  <->  ( n  C_  X  /\  ( ( F `
 z )  i^i 
~P n )  =/=  (/) ) ) )
9 selpw 3958 . . . . . . . . 9  |-  ( n  e.  ~P X  <->  n  C_  X
)
109anbi1i 701 . . . . . . . 8  |-  ( ( n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) )  <->  ( n  C_  X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) )
118, 10syl6bbr 267 . . . . . . 7  |-  ( (
ph  /\  z  e.  X )  ->  (
n  e.  ( ( nei `  J ) `
 { z } )  <->  ( n  e. 
~P X  /\  (
( F `  z
)  i^i  ~P n
)  =/=  (/) ) ) )
1211abbi2dv 2570 . . . . . 6  |-  ( (
ph  /\  z  e.  X )  ->  (
( nei `  J
) `  { z } )  =  {
n  |  ( n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) } )
13 df-rab 2746 . . . . . 6  |-  { n  e.  ~P X  |  ( ( F `  z
)  i^i  ~P n
)  =/=  (/) }  =  { n  |  (
n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) }
1412, 13syl6eqr 2503 . . . . 5  |-  ( (
ph  /\  z  e.  X )  ->  (
( nei `  J
) `  { z } )  =  {
n  e.  ~P X  |  ( ( F `
 z )  i^i 
~P n )  =/=  (/) } )
1514ralrimiva 2802 . . . 4  |-  ( ph  ->  A. z  e.  X  ( ( nei `  J
) `  { z } )  =  {
n  e.  ~P X  |  ( ( F `
 z )  i^i 
~P n )  =/=  (/) } )
16 sneq 3978 . . . . . . 7  |-  ( x  =  z  ->  { x }  =  { z } )
1716fveq2d 5869 . . . . . 6  |-  ( x  =  z  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { z } ) )
18 fveq2 5865 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1918ineq1d 3633 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  i^i  ~P n
)  =  ( ( F `  z )  i^i  ~P n ) )
2019neeq1d 2683 . . . . . . 7  |-  ( x  =  z  ->  (
( ( F `  x )  i^i  ~P n )  =/=  (/)  <->  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) )
2120rabbidv 3036 . . . . . 6  |-  ( x  =  z  ->  { n  e.  ~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) }  =  { n  e.  ~P X  |  ( ( F `  z )  i^i  ~P n )  =/=  (/) } )
2217, 21eqeq12d 2466 . . . . 5  |-  ( x  =  z  ->  (
( ( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  ( ( nei `  J ) `  {
z } )  =  { n  e.  ~P X  |  ( ( F `  z )  i^i  ~P n )  =/=  (/) } ) )
2322cbvralv 3019 . . . 4  |-  ( A. x  e.  X  (
( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  A. z  e.  X  ( ( nei `  J
) `  { z } )  =  {
n  e.  ~P X  |  ( ( F `
 z )  i^i 
~P n )  =/=  (/) } )
2415, 23sylibr 216 . . 3  |-  ( ph  ->  A. x  e.  X  ( ( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )
25 toponuni 19942 . . . . . . . . . 10  |-  ( j  e.  (TopOn `  X
)  ->  X  =  U. j )
26 eqimss2 3485 . . . . . . . . . 10  |-  ( X  =  U. j  ->  U. j  C_  X )
2725, 26syl 17 . . . . . . . . 9  |-  ( j  e.  (TopOn `  X
)  ->  U. j  C_  X )
28 sspwuni 4367 . . . . . . . . 9  |-  ( j 
C_  ~P X  <->  U. j  C_  X )
2927, 28sylibr 216 . . . . . . . 8  |-  ( j  e.  (TopOn `  X
)  ->  j  C_  ~P X )
3029ad2antlr 733 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  ->  j  C_  ~P X )
31 dfss1 3637 . . . . . . 7  |-  ( j 
C_  ~P X  <->  ( ~P X  i^i  j )  =  j )
3230, 31sylib 200 . . . . . 6  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  ->  ( ~P X  i^i  j )  =  j )
33 topontop 19941 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  X
)  ->  j  e.  Top )
3433ad3antlr 737 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  /\  o  e.  ~P X )  -> 
j  e.  Top )
35 eltop2 19991 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (
o  e.  j  <->  A. x  e.  o  E. z  e.  j  ( x  e.  z  /\  z  C_  o ) ) )
3634, 35syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  /\  o  e.  ~P X )  -> 
( o  e.  j  <->  A. x  e.  o  E. z  e.  j 
( x  e.  z  /\  z  C_  o
) ) )
37 elpwi 3960 . . . . . . . . . . . . . . 15  |-  ( o  e.  ~P X  -> 
o  C_  X )
38 ssralv 3493 . . . . . . . . . . . . . . 15  |-  ( o 
C_  X  ->  ( A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  ->  A. x  e.  o  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } ) )
3937, 38syl 17 . . . . . . . . . . . . . 14  |-  ( o  e.  ~P X  -> 
( A. x  e.  X  ( ( nei `  j ) `  {
x } )  =  { n  e.  ~P X  |  ( ( F `  x )  i^i  ~P n )  =/=  (/) }  ->  A. x  e.  o  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } ) )
4039adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  e.  ~P X )  ->  ( A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  ->  A. x  e.  o  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } ) )
41 simprr 766 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )
4241eleq2d 2514 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
( o  e.  ( ( nei `  j
) `  { x } )  <->  o  e.  { n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )
4333ad3antlr 737 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
j  e.  Top )
4425adantl 468 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  X  =  U. j )
4544sseq2d 3460 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( o  C_  X 
<->  o  C_  U. j
) )
4645biimpa 487 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  C_  X )  ->  o  C_ 
U. j )
4737, 46sylan2 477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  e.  ~P X )  ->  o  C_ 
U. j )
4847sselda 3432 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  x  e.  o )  ->  x  e.  U. j
)
4948adantrr 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  ->  x  e.  U. j
)
5047adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
o  C_  U. j
)
51 eqid 2451 . . . . . . . . . . . . . . . . . . 19  |-  U. j  =  U. j
5251isneip 20121 . . . . . . . . . . . . . . . . . 18  |-  ( ( j  e.  Top  /\  x  e.  U. j
)  ->  ( o  e.  ( ( nei `  j
) `  { x } )  <->  ( o  C_ 
U. j  /\  E. z  e.  j  (
x  e.  z  /\  z  C_  o ) ) ) )
5352baibd 920 . . . . . . . . . . . . . . . . 17  |-  ( ( ( j  e.  Top  /\  x  e.  U. j
)  /\  o  C_  U. j )  ->  (
o  e.  ( ( nei `  j ) `
 { x }
)  <->  E. z  e.  j  ( x  e.  z  /\  z  C_  o
) ) )
5443, 49, 50, 53syl21anc 1267 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
( o  e.  ( ( nei `  j
) `  { x } )  <->  E. z  e.  j  ( x  e.  z  /\  z  C_  o ) ) )
55 pweq 3954 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  o  ->  ~P n  =  ~P o
)
5655ineq2d 3634 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  o  ->  (
( F `  x
)  i^i  ~P n
)  =  ( ( F `  x )  i^i  ~P o ) )
5756neeq1d 2683 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  o  ->  (
( ( F `  x )  i^i  ~P n )  =/=  (/)  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
5857elrab3 3197 . . . . . . . . . . . . . . . . 17  |-  ( o  e.  ~P X  -> 
( o  e.  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  ( ( F `
 x )  i^i 
~P o )  =/=  (/) ) )
5958ad2antlr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
( o  e.  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  ( ( F `
 x )  i^i 
~P o )  =/=  (/) ) )
6042, 54, 593bitr3d 287 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  ( x  e.  o  /\  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )  -> 
( E. z  e.  j  ( x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
6160expr 620 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  x  e.  o )  ->  ( ( ( nei `  j ) `  {
x } )  =  { n  e.  ~P X  |  ( ( F `  x )  i^i  ~P n )  =/=  (/) }  ->  ( E. z  e.  j  (
x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) ) )
6261ralimdva 2796 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  e.  ~P X )  ->  ( A. x  e.  o 
( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  ->  A. x  e.  o  ( E. z  e.  j  (
x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) ) )
6340, 62syld 45 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  e.  ~P X )  ->  ( A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  ->  A. x  e.  o  ( E. z  e.  j  (
x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) ) )
6463imp 431 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  ->  A. x  e.  o  ( E. z  e.  j  (
x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
6564an32s 813 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  /\  o  e.  ~P X )  ->  A. x  e.  o 
( E. z  e.  j  ( x  e.  z  /\  z  C_  o )  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
66 ralbi 2921 . . . . . . . . . 10  |-  ( A. x  e.  o  ( E. z  e.  j 
( x  e.  z  /\  z  C_  o
)  <->  ( ( F `
 x )  i^i 
~P o )  =/=  (/) )  ->  ( A. x  e.  o  E. z  e.  j  (
x  e.  z  /\  z  C_  o )  <->  A. x  e.  o  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
6765, 66syl 17 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  /\  o  e.  ~P X )  -> 
( A. x  e.  o  E. z  e.  j  ( x  e.  z  /\  z  C_  o )  <->  A. x  e.  o  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
6836, 67bitrd 257 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  /\  o  e.  ~P X )  -> 
( o  e.  j  <->  A. x  e.  o 
( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
6968rabbi2dva 3640 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  ->  ( ~P X  i^i  j )  =  {
o  e.  ~P X  |  A. x  e.  o  ( ( F `  x )  i^i  ~P o )  =/=  (/) } )
7069, 4syl6eqr 2503 . . . . . 6  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  ->  ( ~P X  i^i  j )  =  J )
7132, 70eqtr3d 2487 . . . . 5  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  ->  j  =  J )
7271expl 624 . . . 4  |-  ( ph  ->  ( ( j  e.  (TopOn `  X )  /\  A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  ->  j  =  J ) )
7372alrimiv 1773 . . 3  |-  ( ph  ->  A. j ( ( j  e.  (TopOn `  X )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  ->  j  =  J ) )
74 eleq1 2517 . . . . 5  |-  ( j  =  J  ->  (
j  e.  (TopOn `  X )  <->  J  e.  (TopOn `  X ) ) )
75 fveq2 5865 . . . . . . . 8  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
7675fveq1d 5867 . . . . . . 7  |-  ( j  =  J  ->  (
( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
7776eqeq1d 2453 . . . . . 6  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  ( ( nei `  J ) `  {
x } )  =  { n  e.  ~P X  |  ( ( F `  x )  i^i  ~P n )  =/=  (/) } ) )
7877ralbidv 2827 . . . . 5  |-  ( j  =  J  ->  ( A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  A. x  e.  X  ( ( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )
7974, 78anbi12d 717 . . . 4  |-  ( j  =  J  ->  (
( j  e.  (TopOn `  X )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  <->  ( J  e.  (TopOn `  X )  /\  A. x  e.  X  ( ( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) ) )
8079eqeu 3209 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  ( J  e.  (TopOn `  X
)  /\  A. x  e.  X  ( ( nei `  J ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )  /\  A. j ( ( j  e.  (TopOn `  X )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )  ->  j  =  J ) )  ->  E! j ( j  e.  (TopOn `  X )  /\  A. x  e.  X  ( ( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )
815, 5, 24, 73, 80syl121anc 1273 . 2  |-  ( ph  ->  E! j ( j  e.  (TopOn `  X
)  /\  A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } ) )
82 df-reu 2744 . 2  |-  ( E! j  e.  (TopOn `  X ) A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) }  <->  E! j
( j  e.  (TopOn `  X )  /\  A. x  e.  X  (
( nei `  j
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } ) )
8381, 82sylibr 216 1  |-  ( ph  ->  E! j  e.  (TopOn `  X ) A. x  e.  X  ( ( nei `  j ) `  { x } )  =  { n  e. 
~P X  |  ( ( F `  x
)  i^i  ~P n
)  =/=  (/) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985   A.wal 1442    = wceq 1444    e. wcel 1887   E!weu 2299   {cab 2437    =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739   {crab 2741    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   U.cuni 4198   -->wf 5578   ` cfv 5582   Topctop 19917  TopOnctopon 19918   neicnei 20113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-topgen 15342  df-top 19921  df-topon 19923  df-nei 20114
This theorem is referenced by: (None)
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