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Theorem neibastop3 30385
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 30382 . . 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 30384 . . . . . . . 8  |-  ( (
ph  /\  z  e.  X )  ->  (
n  e.  ( ( nei `  J ) `
 { z } )  <->  ( n  C_  X  /\  ( ( F `
 z )  i^i 
~P n )  =/=  (/) ) ) )
9 selpw 4022 . . . . . . . . 9  |-  ( n  e.  ~P X  <->  n  C_  X
)
109anbi1i 695 . . . . . . . 8  |-  ( ( n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) )  <->  ( n  C_  X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) )
118, 10syl6bbr 263 . . . . . . 7  |-  ( (
ph  /\  z  e.  X )  ->  (
n  e.  ( ( nei `  J ) `
 { z } )  <->  ( n  e. 
~P X  /\  (
( F `  z
)  i^i  ~P n
)  =/=  (/) ) ) )
1211abbi2dv 2594 . . . . . 6  |-  ( (
ph  /\  z  e.  X )  ->  (
( nei `  J
) `  { z } )  =  {
n  |  ( n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) } )
13 df-rab 2816 . . . . . 6  |-  { n  e.  ~P X  |  ( ( F `  z
)  i^i  ~P n
)  =/=  (/) }  =  { n  |  (
n  e.  ~P X  /\  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) }
1412, 13syl6eqr 2516 . . . . 5  |-  ( (
ph  /\  z  e.  X )  ->  (
( nei `  J
) `  { z } )  =  {
n  e.  ~P X  |  ( ( F `
 z )  i^i 
~P n )  =/=  (/) } )
1514ralrimiva 2871 . . . 4  |-  ( ph  ->  A. z  e.  X  ( ( nei `  J
) `  { z } )  =  {
n  e.  ~P X  |  ( ( F `
 z )  i^i 
~P n )  =/=  (/) } )
16 sneq 4042 . . . . . . 7  |-  ( x  =  z  ->  { x }  =  { z } )
1716fveq2d 5876 . . . . . 6  |-  ( x  =  z  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { z } ) )
18 fveq2 5872 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1918ineq1d 3695 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  i^i  ~P n
)  =  ( ( F `  z )  i^i  ~P n ) )
2019neeq1d 2734 . . . . . . 7  |-  ( x  =  z  ->  (
( ( F `  x )  i^i  ~P n )  =/=  (/)  <->  ( ( F `  z )  i^i  ~P n )  =/=  (/) ) )
2120rabbidv 3101 . . . . . 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 2479 . . . . 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 3084 . . . 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 212 . . 3  |-  ( ph  ->  A. x  e.  X  ( ( nei `  J
) `  { x } )  =  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) } )
25 toponuni 19555 . . . . . . . . . 10  |-  ( j  e.  (TopOn `  X
)  ->  X  =  U. j )
26 eqimss2 3552 . . . . . . . . . 10  |-  ( X  =  U. j  ->  U. j  C_  X )
2725, 26syl 16 . . . . . . . . 9  |-  ( j  e.  (TopOn `  X
)  ->  U. j  C_  X )
28 sspwuni 4421 . . . . . . . . 9  |-  ( j 
C_  ~P X  <->  U. j  C_  X )
2927, 28sylibr 212 . . . . . . . 8  |-  ( j  e.  (TopOn `  X
)  ->  j  C_  ~P X )
3029ad2antlr 726 . . . . . . 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 3699 . . . . . . 7  |-  ( j 
C_  ~P X  <->  ( ~P X  i^i  j )  =  j )
3230, 31sylib 196 . . . . . 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 19554 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  X
)  ->  j  e.  Top )
3433ad3antlr 730 . . . . . . . . . 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 19604 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (
o  e.  j  <->  A. x  e.  o  E. z  e.  j  ( x  e.  z  /\  z  C_  o ) ) )
3634, 35syl 16 . . . . . . . . 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 4024 . . . . . . . . . . . . . . 15  |-  ( o  e.  ~P X  -> 
o  C_  X )
38 ssralv 3560 . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  X  =  U. j )
4544sseq2d 3527 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( o  C_  X 
<->  o  C_  U. j
) )
4645biimpa 484 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  C_  X )  ->  o  C_ 
U. j )
4737, 46sylan2 474 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  o  e.  ~P X )  ->  o  C_ 
U. j )
4847sselda 3499 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  o  e.  ~P X )  /\  x  e.  o )  ->  x  e.  U. j
)
4948adantrr 716 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . 19  |-  U. j  =  U. j
5251isneip 19733 . . . . . . . . . . . . . . . . . 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 909 . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  o  ->  ~P n  =  ~P o
)
5655ineq2d 3696 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  o  ->  (
( F `  x
)  i^i  ~P n
)  =  ( ( F `  x )  i^i  ~P o ) )
5756neeq1d 2734 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  o  ->  (
( ( F `  x )  i^i  ~P n )  =/=  (/)  <->  ( ( F `  x )  i^i  ~P o )  =/=  (/) ) )
5857elrab3 3258 . . . . . . . . . . . . . . . . 17  |-  ( o  e.  ~P X  -> 
( o  e.  {
n  e.  ~P X  |  ( ( F `
 x )  i^i 
~P n )  =/=  (/) }  <->  ( ( F `
 x )  i^i 
~P o )  =/=  (/) ) )
5958ad2antlr 726 . . . . . . . . . . . . . . . 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 283 . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . 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 2865 . . . . . . . . . . . . 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 44 . . . . . . . . . . . 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 429 . . . . . . . . . . 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 804 . . . . . . . . . 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 2988 . . . . . . . . . 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 16 . . . . . . . . 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 253 . . . . . . . 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 3702 . . . . . . 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 2516 . . . . . 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 2500 . . . . 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 618 . . . 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 1720 . . 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 2529 . . . . 5  |-  ( j  =  J  ->  (
j  e.  (TopOn `  X )  <->  J  e.  (TopOn `  X ) ) )
75 fveq2 5872 . . . . . . . 8  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
7675fveq1d 5874 . . . . . . 7  |-  ( j  =  J  ->  (
( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
7776eqeq1d 2459 . . . . . 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 2896 . . . . 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 710 . . . 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 3270 . . 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 1233 . 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 2814 . 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 212 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 184    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395    e. wcel 1819   E!weu 2283   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   E!wreu 2809   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U.cuni 4251   -->wf 5590   ` cfv 5594   Topctop 19521  TopOnctopon 19522   neicnei 19725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-topgen 14861  df-top 19526  df-topon 19529  df-nei 19726
This theorem is referenced by: (None)
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