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Theorem kqnrmlem1 17777
Description: A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqnrmlem1  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqnrmlem1
Dummy variables  m  w  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqtopon 17761 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
32adantr 453 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  (TopOn `  ran  F ) )
4 topontop 16993 . . 3  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
53, 4syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Top )
6 simplr 733 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  J  e.  Nrm )
71kqid 17762 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
87ad2antrr 708 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
9 simprl 734 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  z  e.  (KQ `  J ) )
10 cnima 17331 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  z  e.  (KQ `  J ) )  ->  ( `' F " z )  e.  J )
118, 9, 10syl2anc 644 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " z )  e.  J )
12 inss1 3563 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ( Clsd `  (KQ `  J
) )
13 simprr 735 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) )
1412, 13sseldi 3348 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ( Clsd `  (KQ `  J
) ) )
15 cnclima 17334 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  w  e.  ( Clsd `  (KQ `  J ) ) )  ->  ( `' F " w )  e.  (
Clsd `  J )
)
168, 14, 15syl2anc 644 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " w )  e.  ( Clsd `  J
) )
17 inss2 3564 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ~P z
1817, 13sseldi 3348 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ~P z )
19 elpwi 3809 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
20 imass2 5242 . . . . . 6  |-  ( w 
C_  z  ->  ( `' F " w ) 
C_  ( `' F " z ) )
2118, 19, 203syl 19 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " w )  C_  ( `' F " z ) )
22 nrmsep3 17421 . . . . 5  |-  ( ( J  e.  Nrm  /\  ( ( `' F " z )  e.  J  /\  ( `' F "
w )  e.  (
Clsd `  J )  /\  ( `' F "
w )  C_  ( `' F " z ) ) )  ->  E. u  e.  J  ( ( `' F " w ) 
C_  u  /\  (
( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
236, 11, 16, 21, 22syl13anc 1187 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  E. u  e.  J  ( ( `' F " w ) 
C_  u  /\  (
( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
24 simplll 736 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  J  e.  (TopOn `  X ) )
25 simprl 734 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  e.  J )
261kqopn 17768 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  u  e.  J )  ->  ( F " u )  e.  (KQ `  J ) )
2724, 25, 26syl2anc 644 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " u )  e.  (KQ
`  J ) )
28 simprrl 742 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( `' F " w )  C_  u )
291kqffn 17759 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
30 fnfun 5544 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3124, 29, 303syl 19 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  Fun  F )
3214adantr 453 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  e.  ( Clsd `  (KQ `  J
) ) )
33 eqid 2438 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
3433cldss 17095 . . . . . . . . 9  |-  ( w  e.  ( Clsd `  (KQ `  J ) )  ->  w  C_  U. (KQ `  J ) )
3532, 34syl 16 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  U. (KQ `  J ) )
36 toponuni 16994 . . . . . . . . 9  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3724, 2, 363syl 19 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ran  F  = 
U. (KQ `  J
) )
3835, 37sseqtr4d 3387 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  ran  F )
39 funimass1 5528 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
ran  F )  -> 
( ( `' F " w )  C_  u  ->  w  C_  ( F " u ) ) )
4031, 38, 39syl2anc 644 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( `' F " w ) 
C_  u  ->  w  C_  ( F " u
) ) )
4128, 40mpd 15 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  ( F " u ) )
42 topontop 16993 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
4324, 42syl 16 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  J  e.  Top )
44 elssuni 4045 . . . . . . . . . 10  |-  ( u  e.  J  ->  u  C_ 
U. J )
4544ad2antrl 710 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  C_  U. J
)
46 eqid 2438 . . . . . . . . . 10  |-  U. J  =  U. J
4746clscld 17113 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  e.  ( Clsd `  J ) )
4843, 45, 47syl2anc 644 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  e.  (
Clsd `  J )
)
491kqcld 17769 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  (
( cls `  J
) `  u )  e.  ( Clsd `  J
) )  ->  ( F " ( ( cls `  J ) `  u
) )  e.  (
Clsd `  (KQ `  J
) ) )
5024, 48, 49syl2anc 644 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " ( ( cls `  J
) `  u )
)  e.  ( Clsd `  (KQ `  J ) ) )
5146sscls 17122 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  u  C_  (
( cls `  J
) `  u )
)
5243, 45, 51syl2anc 644 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  C_  (
( cls `  J
) `  u )
)
53 imass2 5242 . . . . . . . 8  |-  ( u 
C_  ( ( cls `  J ) `  u
)  ->  ( F " u )  C_  ( F " ( ( cls `  J ) `  u
) ) )
5452, 53syl 16 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " u )  C_  ( F " ( ( cls `  J ) `  u
) ) )
5533clsss2 17138 . . . . . . 7  |-  ( ( ( F " (
( cls `  J
) `  u )
)  e.  ( Clsd `  (KQ `  J ) )  /\  ( F
" u )  C_  ( F " ( ( cls `  J ) `
 u ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  ( F "
( ( cls `  J
) `  u )
) )
5650, 54, 55syl2anc 644 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  ( F "
( ( cls `  J
) `  u )
) )
57 simprrr 743 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  ( `' F " z ) )
5846clsss3 17125 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  C_  U. J )
5943, 45, 58syl2anc 644 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  U. J
)
60 fndm 5546 . . . . . . . . . . 11  |-  ( F  Fn  X  ->  dom  F  =  X )
6124, 29, 603syl 19 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  dom  F  =  X )
62 toponuni 16994 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
6324, 62syl 16 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  X  =  U. J )
6461, 63eqtrd 2470 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  dom  F  = 
U. J )
6559, 64sseqtr4d 3387 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  dom  F )
66 funimass3 5848 . . . . . . . 8  |-  ( ( Fun  F  /\  (
( cls `  J
) `  u )  C_ 
dom  F )  -> 
( ( F "
( ( cls `  J
) `  u )
)  C_  z  <->  ( ( cls `  J ) `  u )  C_  ( `' F " z ) ) )
6731, 65, 66syl2anc 644 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( F " ( ( cls `  J ) `  u
) )  C_  z  <->  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
6857, 67mpbird 225 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " ( ( cls `  J
) `  u )
)  C_  z )
6956, 68sstrd 3360 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  z )
70 sseq2 3372 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
w  C_  m  <->  w  C_  ( F " u ) ) )
71 fveq2 5730 . . . . . . . 8  |-  ( m  =  ( F "
u )  ->  (
( cls `  (KQ `  J ) ) `  m )  =  ( ( cls `  (KQ `  J ) ) `  ( F " u ) ) )
7271sseq1d 3377 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
( ( cls `  (KQ `  J ) ) `  m )  C_  z  <->  ( ( cls `  (KQ `  J ) ) `  ( F " u ) )  C_  z )
)
7370, 72anbi12d 693 . . . . . 6  |-  ( m  =  ( F "
u )  ->  (
( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
)  <->  ( w  C_  ( F " u )  /\  ( ( cls `  (KQ `  J ) ) `  ( F
" u ) ) 
C_  z ) ) )
7473rspcev 3054 . . . . 5  |-  ( ( ( F " u
)  e.  (KQ `  J )  /\  (
w  C_  ( F " u )  /\  (
( cls `  (KQ `  J ) ) `  ( F " u ) )  C_  z )
)  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7527, 41, 69, 74syl12anc 1183 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7623, 75rexlimddv 2836 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7776ralrimivva 2800 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  A. z  e.  (KQ `  J ) A. w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) E. m  e.  (KQ `  J ) ( w 
C_  m  /\  (
( cls `  (KQ `  J ) ) `  m )  C_  z
) )
78 isnrm 17401 . 2  |-  ( (KQ
`  J )  e. 
Nrm 
<->  ( (KQ `  J
)  e.  Top  /\  A. z  e.  (KQ `  J ) A. w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) E. m  e.  (KQ `  J ) ( w 
C_  m  /\  (
( cls `  (KQ `  J ) ) `  m )  C_  z
) ) )
795, 77, 78sylanbrc 647 1  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883   Fun wfun 5450    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   Topctop 16960  TopOnctopon 16961   Clsdccld 17082   clsccl 17084    Cn ccn 17290   Nrmcnrm 17376  KQckq 17727
This theorem is referenced by:  kqnrm  17786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-qtop 13735  df-top 16965  df-topon 16968  df-cld 17085  df-cls 17087  df-cn 17293  df-nrm 17383  df-kq 17728
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