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Theorem kqnrmlem1 19338
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 19322 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
32adantr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  (TopOn `  ran  F ) )
4 topontop 18553 . . 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 754 . . . . 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 19323 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
87ad2antrr 725 . . . . . 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 755 . . . . . 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 18891 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  z  e.  (KQ `  J ) )  ->  ( `' F " z )  e.  J )
118, 9, 10syl2anc 661 . . . . 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 3591 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ( Clsd `  (KQ `  J
) )
13 simprr 756 . . . . . . 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 3375 . . . . . 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 18894 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  w  e.  ( Clsd `  (KQ `  J ) ) )  ->  ( `' F " w )  e.  (
Clsd `  J )
)
168, 14, 15syl2anc 661 . . . . 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 3592 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ~P z
1817, 13sseldi 3375 . . . . . 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 3890 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
20 imass2 5225 . . . . . 6  |-  ( w 
C_  z  ->  ( `' F " w ) 
C_  ( `' F " z ) )
2118, 19, 203syl 20 . . . . 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 18981 . . . . 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 1220 . . . 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 757 . . . . . 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 755 . . . . . 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 19329 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  u  e.  J )  ->  ( F " u )  e.  (KQ `  J ) )
2724, 25, 26syl2anc 661 . . . . 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 763 . . . . . 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 19320 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
30 fnfun 5529 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3124, 29, 303syl 20 . . . . . . 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 465 . . . . . . . . 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 2443 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
3433cldss 18655 . . . . . . . . 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 18554 . . . . . . . . 9  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3724, 2, 363syl 20 . . . . . . . 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 3414 . . . . . . 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 5512 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
ran  F )  -> 
( ( `' F " w )  C_  u  ->  w  C_  ( F " u ) ) )
4031, 38, 39syl2anc 661 . . . . . 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 18553 . . . . . . . . . 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 4142 . . . . . . . . . 10  |-  ( u  e.  J  ->  u  C_ 
U. J )
4544ad2antrl 727 . . . . . . . . 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 2443 . . . . . . . . . 10  |-  U. J  =  U. J
4746clscld 18673 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  e.  ( Clsd `  J ) )
4843, 45, 47syl2anc 661 . . . . . . . 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 19330 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  (
( cls `  J
) `  u )  e.  ( Clsd `  J
) )  ->  ( F " ( ( cls `  J ) `  u
) )  e.  (
Clsd `  (KQ `  J
) ) )
5024, 48, 49syl2anc 661 . . . . . . 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 18682 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  u  C_  (
( cls `  J
) `  u )
)
5243, 45, 51syl2anc 661 . . . . . . . 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 5225 . . . . . . . 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 18698 . . . . . . 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 661 . . . . . 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 764 . . . . . . 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 18685 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  C_  U. J )
5943, 45, 58syl2anc 661 . . . . . . . . 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 5531 . . . . . . . . . . 11  |-  ( F  Fn  X  ->  dom  F  =  X )
6124, 29, 603syl 20 . . . . . . . . . 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 18554 . . . . . . . . . . 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 2475 . . . . . . . . 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 3414 . . . . . . . 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 5840 . . . . . . . 8  |-  ( ( Fun  F  /\  (
( cls `  J
) `  u )  C_ 
dom  F )  -> 
( ( F "
( ( cls `  J
) `  u )
)  C_  z  <->  ( ( cls `  J ) `  u )  C_  ( `' F " z ) ) )
6731, 65, 66syl2anc 661 . . . . . . 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 232 . . . . . 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 3387 . . . . 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 3399 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
w  C_  m  <->  w  C_  ( F " u ) ) )
71 fveq2 5712 . . . . . . . 8  |-  ( m  =  ( F "
u )  ->  (
( cls `  (KQ `  J ) ) `  m )  =  ( ( cls `  (KQ `  J ) ) `  ( F " u ) ) )
7271sseq1d 3404 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
( ( cls `  (KQ `  J ) ) `  m )  C_  z  <->  ( ( cls `  (KQ `  J ) ) `  ( F " u ) )  C_  z )
)
7370, 72anbi12d 710 . . . . . 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 3094 . . . . 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 1216 . . . 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 2866 . . 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 2829 . 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 18961 . 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 664 1  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   U.cuni 4112    e. cmpt 4371   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864   Fun wfun 5433    Fn wfn 5434   ` cfv 5439  (class class class)co 6112   Topctop 18520  TopOnctopon 18521   Clsdccld 18642   clsccl 18644    Cn ccn 18850   Nrmcnrm 18936  KQckq 19288
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-qtop 14466  df-top 18525  df-topon 18528  df-cld 18645  df-cls 18647  df-cn 18853  df-nrm 18943  df-kq 19289
This theorem is referenced by:  kqnrm  19347
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