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Theorem kqnrmlem1 20370
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 20354 . . . 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 19554 . . 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 755 . . . . 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 20355 . . . . . . 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 756 . . . . . 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 19893 . . . . . 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 3714 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ( Clsd `  (KQ `  J
) )
13 simprr 757 . . . . . . 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 3497 . . . . . 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 19896 . . . . . 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 3715 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ~P z
1817, 13sseldi 3497 . . . . . 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 4024 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
20 imass2 5382 . . . . . 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 19983 . . . . 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 1230 . . . 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 759 . . . . . 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 756 . . . . . 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 20361 . . . . . 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 765 . . . . . 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 20352 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
30 fnfun 5684 . . . . . . . 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 2457 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
3433cldss 19657 . . . . . . . . 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 19555 . . . . . . . . 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 3536 . . . . . . 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 5667 . . . . . . 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 19554 . . . . . . . . . 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 4281 . . . . . . . . . 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 2457 . . . . . . . . . 10  |-  U. J  =  U. J
4746clscld 19675 . . . . . . . . 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 20362 . . . . . . . 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 19684 . . . . . . . . 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 5382 . . . . . . . 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 19700 . . . . . . 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 766 . . . . . . 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 19687 . . . . . . . . . 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 5686 . . . . . . . . . . 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 19555 . . . . . . . . . . 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 2498 . . . . . . . . 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 3536 . . . . . . . 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 6004 . . . . . . . 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 3509 . . . . 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 3521 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
w  C_  m  <->  w  C_  ( F " u ) ) )
71 fveq2 5872 . . . . . . . 8  |-  ( m  =  ( F "
u )  ->  (
( cls `  (KQ `  J ) ) `  m )  =  ( ( cls `  (KQ `  J ) ) `  ( F " u ) ) )
7271sseq1d 3526 . . . . . . 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 3210 . . . . 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 1226 . . . 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 2953 . . 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 2878 . 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 19963 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   U.cuni 4251    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   Topctop 19521  TopOnctopon 19522   Clsdccld 19644   clsccl 19646    Cn ccn 19852   Nrmcnrm 19938  KQckq 20320
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-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-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-qtop 14924  df-top 19526  df-topon 19529  df-cld 19647  df-cls 19649  df-cn 19855  df-nrm 19945  df-kq 20321
This theorem is referenced by:  kqnrm  20379
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