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Theorem kqnrmlem2 17435
Description: If the Kolmogorov quotient of a space is normal then so is the original space. (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
kqnrmlem2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqnrmlem2
Dummy variables  m  w  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 16664 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 451 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Top )
3 simplr 731 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
(KQ `  J )  e.  Nrm )
4 simpll 730 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  J  e.  (TopOn `  X
) )
5 simprl 732 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
z  e.  J )
6 kqval.2 . . . . . . 7  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
76kqopn 17425 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( F " z )  e.  (KQ `  J ) )
84, 5, 7syl2anc 642 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
( F " z
)  e.  (KQ `  J ) )
9 inss1 3389 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ( Clsd `  J )
10 simprr 733 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  w  e.  ( ( Clsd `  J )  i^i 
~P z ) )
119, 10sseldi 3178 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  w  e.  ( Clsd `  J ) )
126kqcld 17426 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  w  e.  ( Clsd `  J
) )  ->  ( F " w )  e.  ( Clsd `  (KQ `  J ) ) )
134, 11, 12syl2anc 642 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
( F " w
)  e.  ( Clsd `  (KQ `  J ) ) )
14 inss2 3390 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ~P z
1514, 10sseldi 3178 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  w  e.  ~P z
)
16 elpwi 3633 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
17 imass2 5049 . . . . . 6  |-  ( w 
C_  z  ->  ( F " w )  C_  ( F " z ) )
1815, 16, 173syl 18 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
( F " w
)  C_  ( F " z ) )
19 nrmsep3 17083 . . . . 5  |-  ( ( (KQ `  J )  e.  Nrm  /\  (
( F " z
)  e.  (KQ `  J )  /\  ( F " w )  e.  ( Clsd `  (KQ `  J ) )  /\  ( F " w ) 
C_  ( F "
z ) ) )  ->  E. m  e.  (KQ
`  J ) ( ( F " w
)  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z ) ) )
203, 8, 13, 18, 19syl13anc 1184 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  E. m  e.  (KQ `  J ) ( ( F " w ) 
C_  m  /\  (
( cls `  (KQ `  J ) ) `  m )  C_  ( F " z ) ) )
21 simplll 734 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  J  e.  (TopOn `  X ) )
226kqid 17419 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
2321, 22syl 15 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
24 simprl 732 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  m  e.  (KQ `  J ) )
25 cnima 16994 . . . . . . . 8  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  m  e.  (KQ `  J ) )  ->  ( `' F " m )  e.  J )
2623, 24, 25syl2anc 642 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " m )  e.  J )
27 simprrl 740 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( F " w )  C_  m
)
286kqffn 17416 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
29 fnfun 5341 . . . . . . . . . 10  |-  ( F  Fn  X  ->  Fun  F )
3021, 28, 293syl 18 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  Fun  F )
3111adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  w  e.  ( Clsd `  J )
)
32 eqid 2283 . . . . . . . . . . . 12  |-  U. J  =  U. J
3332cldss 16766 . . . . . . . . . . 11  |-  ( w  e.  ( Clsd `  J
)  ->  w  C_  U. J
)
3431, 33syl 15 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  w  C_  U. J
)
35 fndm 5343 . . . . . . . . . . . 12  |-  ( F  Fn  X  ->  dom  F  =  X )
3621, 28, 353syl 18 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  dom  F  =  X )
37 toponuni 16665 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3821, 37syl 15 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  X  =  U. J )
3936, 38eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  dom  F  = 
U. J )
4034, 39sseqtr4d 3215 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  w  C_  dom  F )
41 funimass3 5641 . . . . . . . . 9  |-  ( ( Fun  F  /\  w  C_ 
dom  F )  -> 
( ( F "
w )  C_  m  <->  w 
C_  ( `' F " m ) ) )
4230, 40, 41syl2anc 642 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( ( F " w )  C_  m 
<->  w  C_  ( `' F " m ) ) )
4327, 42mpbid 201 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  w  C_  ( `' F " m ) )
446kqtopon 17418 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
45 topontop 16664 . . . . . . . . . . . 12  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
4621, 44, 453syl 18 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  (KQ `  J
)  e.  Top )
47 elssuni 3855 . . . . . . . . . . . 12  |-  ( m  e.  (KQ `  J
)  ->  m  C_  U. (KQ `  J ) )
4847ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  m  C_  U. (KQ `  J ) )
49 eqid 2283 . . . . . . . . . . . 12  |-  U. (KQ `  J )  =  U. (KQ `  J )
5049clscld 16784 . . . . . . . . . . 11  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  (
( cls `  (KQ `  J ) ) `  m )  e.  (
Clsd `  (KQ `  J
) ) )
5146, 48, 50syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  m
)  e.  ( Clsd `  (KQ `  J ) ) )
52 cnclima 16997 . . . . . . . . . 10  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  (
( cls `  (KQ `  J ) ) `  m )  e.  (
Clsd `  (KQ `  J
) ) )  -> 
( `' F "
( ( cls `  (KQ `  J ) ) `  m ) )  e.  ( Clsd `  J
) )
5323, 51, 52syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) )  e.  ( Clsd `  J ) )
5449sscls 16793 . . . . . . . . . . 11  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  m  C_  ( ( cls `  (KQ `  J ) ) `  m ) )
5546, 48, 54syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  m  C_  (
( cls `  (KQ `  J ) ) `  m ) )
56 imass2 5049 . . . . . . . . . 10  |-  ( m 
C_  ( ( cls `  (KQ `  J ) ) `  m )  ->  ( `' F " m )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5755, 56syl 15 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " m )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5832clsss2 16809 . . . . . . . . 9  |-  ( ( ( `' F "
( ( cls `  (KQ `  J ) ) `  m ) )  e.  ( Clsd `  J
)  /\  ( `' F " m )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )  ->  ( ( cls `  J ) `  ( `' F " m ) )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5953, 57, 58syl2anc 642 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( ( cls `  J ) `  ( `' F " m ) )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
60 simprrr 741 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  m
)  C_  ( F " z ) )
61 imass2 5049 . . . . . . . . . 10  |-  ( ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z )  -> 
( `' F "
( ( cls `  (KQ `  J ) ) `  m ) )  C_  ( `' F " ( F
" z ) ) )
6260, 61syl 15 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) )  C_  ( `' F " ( F "
z ) ) )
635adantr 451 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  z  e.  J )
646kqsat 17422 . . . . . . . . . 10  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( `' F " ( F
" z ) )  =  z )
6521, 63, 64syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " ( F "
z ) )  =  z )
6662, 65sseqtrd 3214 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) )  C_  z )
6759, 66sstrd 3189 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
68 sseq2 3200 . . . . . . . . 9  |-  ( u  =  ( `' F " m )  ->  (
w  C_  u  <->  w  C_  ( `' F " m ) ) )
69 fveq2 5525 . . . . . . . . . 10  |-  ( u  =  ( `' F " m )  ->  (
( cls `  J
) `  u )  =  ( ( cls `  J ) `  ( `' F " m ) ) )
7069sseq1d 3205 . . . . . . . . 9  |-  ( u  =  ( `' F " m )  ->  (
( ( cls `  J
) `  u )  C_  z  <->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
)
7168, 70anbi12d 691 . . . . . . . 8  |-  ( u  =  ( `' F " m )  ->  (
( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z )  <->  ( w  C_  ( `' F "
m )  /\  (
( cls `  J
) `  ( `' F " m ) ) 
C_  z ) ) )
7271rspcev 2884 . . . . . . 7  |-  ( ( ( `' F "
m )  e.  J  /\  ( w  C_  ( `' F " m )  /\  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
)  ->  E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J ) `
 u )  C_  z ) )
7326, 43, 67, 72syl12anc 1180 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  ( m  e.  (KQ `  J
)  /\  ( ( F " w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m ) 
C_  ( F "
z ) ) ) )  ->  E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J ) `
 u )  C_  z ) )
7473expr 598 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  (
( Clsd `  J )  i^i  ~P z ) ) )  /\  m  e.  (KQ `  J ) )  ->  ( (
( F " w
)  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z ) )  ->  E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z ) ) )
7574rexlimdva 2667 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  -> 
( E. m  e.  (KQ `  J ) ( ( F "
w )  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z ) )  ->  E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z ) ) )
7620, 75mpd 14 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  /\  ( z  e.  J  /\  w  e.  ( ( Clsd `  J
)  i^i  ~P z
) ) )  ->  E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z ) )
7776ralrimivva 2635 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  A. z  e.  J  A. w  e.  (
( Clsd `  J )  i^i  ~P z ) E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z ) )
78 isnrm 17063 . 2  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. z  e.  J  A. w  e.  ( ( Clsd `  J
)  i^i  ~P z
) E. u  e.  J  ( w  C_  u  /\  ( ( cls `  J ) `  u
)  C_  z )
) )
792, 77, 78sylanbrc 645 1  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   Clsdccld 16753   clsccl 16755    Cn ccn 16954   Nrmcnrm 17038  KQckq 17384
This theorem is referenced by:  kqnrm  17443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957  df-nrm 17045  df-kq 17385
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