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Theorem kqnrmlem2 20008
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 19222 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 465 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Top )
3 simplr 754 . . . . 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 753 . . . . . 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 755 . . . . . 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 19998 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( F " z )  e.  (KQ `  J ) )
84, 5, 7syl2anc 661 . . . . 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 3718 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ( Clsd `  J )
10 simprr 756 . . . . . . 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 3502 . . . . . 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 19999 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  w  e.  ( Clsd `  J
) )  ->  ( F " w )  e.  ( Clsd `  (KQ `  J ) ) )
134, 11, 12syl2anc 661 . . . . 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 3719 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ~P z
1514, 10sseldi 3502 . . . . . 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 4019 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
17 imass2 5372 . . . . . 6  |-  ( w 
C_  z  ->  ( F " w )  C_  ( F " z ) )
1815, 16, 173syl 20 . . . . 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 19650 . . . . 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 1230 . . . 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 757 . . . . . . 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 ) ) ) )  ->  J  e.  (TopOn `  X ) )
226kqid 19992 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
2321, 22syl 16 . . . . . 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 ) ) ) )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
24 simprl 755 . . . . . 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 ) ) ) )  ->  m  e.  (KQ `  J ) )
25 cnima 19560 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  m  e.  (KQ `  J ) )  ->  ( `' F " m )  e.  J )
2623, 24, 25syl2anc 661 . . . . 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 ) ) ) )  ->  ( `' F " m )  e.  J )
27 simprrl 763 . . . . . 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 ) ) ) )  ->  ( F " w )  C_  m
)
286kqffn 19989 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
29 fnfun 5678 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3021, 28, 293syl 20 . . . . . . 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 ) ) ) )  ->  Fun  F )
3111adantr 465 . . . . . . . . 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  e.  ( Clsd `  J )
)
32 eqid 2467 . . . . . . . . . 10  |-  U. J  =  U. J
3332cldss 19324 . . . . . . . . 9  |-  ( w  e.  ( Clsd `  J
)  ->  w  C_  U. J
)
3431, 33syl 16 . . . . . . . 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 ) ) ) )  ->  w  C_  U. J
)
35 fndm 5680 . . . . . . . . . 10  |-  ( F  Fn  X  ->  dom  F  =  X )
3621, 28, 353syl 20 . . . . . . . . 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 ) ) ) )  ->  dom  F  =  X )
37 toponuni 19223 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3821, 37syl 16 . . . . . . . . 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 ) ) ) )  ->  X  =  U. J )
3936, 38eqtrd 2508 . . . . . . . 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 ) ) ) )  ->  dom  F  = 
U. J )
4034, 39sseqtr4d 3541 . . . . . . 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_  dom  F )
41 funimass3 5997 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
dom  F )  -> 
( ( F "
w )  C_  m  <->  w 
C_  ( `' F " m ) ) )
4230, 40, 41syl2anc 661 . . . . . 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 ) ) ) )  ->  ( ( F " w )  C_  m 
<->  w  C_  ( `' F " m ) ) )
4327, 42mpbid 210 . . . . 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 ) ) ) )  ->  w  C_  ( `' F " m ) )
446kqtopon 19991 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
45 topontop 19222 . . . . . . . . . 10  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
4621, 44, 453syl 20 . . . . . . . . 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 ) ) ) )  ->  (KQ `  J
)  e.  Top )
47 elssuni 4275 . . . . . . . . . 10  |-  ( m  e.  (KQ `  J
)  ->  m  C_  U. (KQ `  J ) )
4847ad2antrl 727 . . . . . . . . 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 ) ) ) )  ->  m  C_  U. (KQ `  J ) )
49 eqid 2467 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
5049clscld 19342 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  (
( cls `  (KQ `  J ) ) `  m )  e.  (
Clsd `  (KQ `  J
) ) )
5146, 48, 50syl2anc 661 . . . . . . . 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 `  (KQ `  J
) ) `  m
)  e.  ( Clsd `  (KQ `  J ) ) )
52 cnclima 19563 . . . . . . . 8  |-  ( ( 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 661 . . . . . . 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 " ( ( cls `  (KQ `  J ) ) `  m ) )  e.  ( Clsd `  J ) )
5449sscls 19351 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  m  C_  ( ( cls `  (KQ `  J ) ) `  m ) )
5546, 48, 54syl2anc 661 . . . . . . . 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  C_  (
( cls `  (KQ `  J ) ) `  m ) )
56 imass2 5372 . . . . . . . 8  |-  ( m 
C_  ( ( cls `  (KQ `  J ) ) `  m )  ->  ( `' F " m )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5755, 56syl 16 . . . . . . 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 )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5832clsss2 19367 . . . . . . 7  |-  ( ( ( `' 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 661 . . . . . 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 ) ) ) )  ->  ( ( cls `  J ) `  ( `' F " m ) )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
60 simprrr 764 . . . . . . . 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 `  (KQ `  J
) ) `  m
)  C_  ( F " z ) )
61 imass2 5372 . . . . . . . 8  |-  ( ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z )  -> 
( `' F "
( ( cls `  (KQ `  J ) ) `  m ) )  C_  ( `' F " ( F
" z ) ) )
6260, 61syl 16 . . . . . . 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 " ( ( cls `  (KQ `  J ) ) `  m ) )  C_  ( `' F " ( F "
z ) ) )
635adantr 465 . . . . . . . 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 ) ) ) )  ->  z  e.  J )
646kqsat 19995 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( `' F " ( F
" z ) )  =  z )
6521, 63, 64syl2anc 661 . . . . . . 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 " ( F "
z ) )  =  z )
6662, 65sseqtrd 3540 . . . . . 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 ) ) ) )  ->  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) )  C_  z )
6759, 66sstrd 3514 . . . . 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 ) ) ) )  ->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
68 sseq2 3526 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
w  C_  u  <->  w  C_  ( `' F " m ) ) )
69 fveq2 5866 . . . . . . . 8  |-  ( u  =  ( `' F " m )  ->  (
( cls `  J
) `  u )  =  ( ( cls `  J ) `  ( `' F " m ) ) )
7069sseq1d 3531 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
( ( cls `  J
) `  u )  C_  z  <->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
)
7168, 70anbi12d 710 . . . . . 6  |-  ( u  =  ( `' F " m )  ->  (
( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z )  <->  ( w  C_  ( `' F "
m )  /\  (
( cls `  J
) `  ( `' F " m ) ) 
C_  z ) ) )
7271rspcev 3214 . . . . 5  |-  ( ( ( `' 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 1226 . . . 4  |-  ( ( ( ( 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 ) )
7420, 73rexlimddv 2959 . . 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 ) )
7574ralrimivva 2885 . 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 ) )
76 isnrm 19630 . 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 )
) )
772, 75, 76sylanbrc 664 1  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5582    Fn wfn 5583   ` cfv 5588  (class class class)co 6284   Topctop 19189  TopOnctopon 19190   Clsdccld 19311   clsccl 19313    Cn ccn 19519   Nrmcnrm 19605  KQckq 19957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-qtop 14762  df-top 19194  df-topon 19197  df-cld 19314  df-cls 19316  df-cn 19522  df-nrm 19612  df-kq 19958
This theorem is referenced by:  kqnrm  20016
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