MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqnrmlem2 Unicode version

Theorem kqnrmlem2 17729
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 16946 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 452 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Top )
3 simplr 732 . . . . 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 731 . . . . . 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 733 . . . . . 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 17719 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( F " z )  e.  (KQ `  J ) )
84, 5, 7syl2anc 643 . . . . 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 3521 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ( Clsd `  J )
10 simprr 734 . . . . . . 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 3306 . . . . . 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 17720 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  w  e.  ( Clsd `  J
) )  ->  ( F " w )  e.  ( Clsd `  (KQ `  J ) ) )
134, 11, 12syl2anc 643 . . . . 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 3522 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ~P z
1514, 10sseldi 3306 . . . . . 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 3767 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
17 imass2 5199 . . . . . 6  |-  ( w 
C_  z  ->  ( F " w )  C_  ( F " z ) )
1815, 16, 173syl 19 . . . . 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 17373 . . . . 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 1186 . . . 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 735 . . . . . . 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 17713 . . . . . . 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 733 . . . . . 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 17283 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  m  e.  (KQ `  J ) )  ->  ( `' F " m )  e.  J )
2623, 24, 25syl2anc 643 . . . . 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 741 . . . . . 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 17710 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
29 fnfun 5501 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3021, 28, 293syl 19 . . . . . . 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 452 . . . . . . . . 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 2404 . . . . . . . . . 10  |-  U. J  =  U. J
3332cldss 17048 . . . . . . . . 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 5503 . . . . . . . . . 10  |-  ( F  Fn  X  ->  dom  F  =  X )
3621, 28, 353syl 19 . . . . . . . . 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 16947 . . . . . . . . . 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 2436 . . . . . . . 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 3345 . . . . . . 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 5805 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
dom  F )  -> 
( ( F "
w )  C_  m  <->  w 
C_  ( `' F " m ) ) )
4230, 40, 41syl2anc 643 . . . . . 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 202 . . . . 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 17712 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
45 topontop 16946 . . . . . . . . . 10  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
4621, 44, 453syl 19 . . . . . . . . 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 4003 . . . . . . . . . 10  |-  ( m  e.  (KQ `  J
)  ->  m  C_  U. (KQ `  J ) )
4847ad2antrl 709 . . . . . . . . 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 2404 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
5049clscld 17066 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  (
( cls `  (KQ `  J ) ) `  m )  e.  (
Clsd `  (KQ `  J
) ) )
5146, 48, 50syl2anc 643 . . . . . . . 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 17286 . . . . . . . 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 643 . . . . . . 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 17075 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  m  C_  ( ( cls `  (KQ `  J ) ) `  m ) )
5546, 48, 54syl2anc 643 . . . . . . . 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 5199 . . . . . . . 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 17091 . . . . . . 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 643 . . . . . 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 742 . . . . . . . 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 5199 . . . . . . . 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 452 . . . . . . . 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 17716 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( `' F " ( F
" z ) )  =  z )
6521, 63, 64syl2anc 643 . . . . . . 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 3344 . . . . . 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 3318 . . . . 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 3330 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
w  C_  u  <->  w  C_  ( `' F " m ) ) )
69 fveq2 5687 . . . . . . . 8  |-  ( u  =  ( `' F " m )  ->  (
( cls `  J
) `  u )  =  ( ( cls `  J ) `  ( `' F " m ) ) )
7069sseq1d 3335 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
( ( cls `  J
) `  u )  C_  z  <->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
)
7168, 70anbi12d 692 . . . . . 6  |-  ( u  =  ( `' F " m )  ->  (
( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z )  <->  ( w  C_  ( `' F "
m )  /\  (
( cls `  J
) `  ( `' F " m ) ) 
C_  z ) ) )
7271rspcev 3012 . . . . 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 1182 . . . 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 2794 . . 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 2758 . 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 17353 . 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 646 1  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Topctop 16913  TopOnctopon 16914   Clsdccld 17035   clsccl 17037    Cn ccn 17242   Nrmcnrm 17328  KQckq 17678
This theorem is referenced by:  kqnrm  17737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-qtop 13688  df-top 16918  df-topon 16921  df-cld 17038  df-cls 17040  df-cn 17245  df-nrm 17335  df-kq 17679
  Copyright terms: Public domain W3C validator