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

Theorem kqnrmlem2 20745
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 19927 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
21adantr 466 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Top )
3 simplr 760 . . . . 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 758 . . . . . 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 762 . . . . . 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 20735 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( F " z )  e.  (KQ `  J ) )
84, 5, 7syl2anc 665 . . . . 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 3682 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ( Clsd `  J )
10 simprr 764 . . . . . . 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 3462 . . . . . 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 20736 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  w  e.  ( Clsd `  J
) )  ->  ( F " w )  e.  ( Clsd `  (KQ `  J ) ) )
134, 11, 12syl2anc 665 . . . . 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 3683 . . . . . . 7  |-  ( (
Clsd `  J )  i^i  ~P z )  C_  ~P z
1514, 10sseldi 3462 . . . . . 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 3988 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
17 imass2 5219 . . . . . 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 20357 . . . . 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 1266 . . . 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 766 . . . . . . 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 20729 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
2321, 22syl 17 . . . . . 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 762 . . . . . 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 20267 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  m  e.  (KQ `  J ) )  ->  ( `' F " m )  e.  J )
2623, 24, 25syl2anc 665 . . . . 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 772 . . . . . 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 20726 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
29 fnfun 5687 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3021, 28, 293syl 18 . . . . . . 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 466 . . . . . . . . 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 2422 . . . . . . . . . 10  |-  U. J  =  U. J
3332cldss 20030 . . . . . . . . 9  |-  ( w  e.  ( Clsd `  J
)  ->  w  C_  U. J
)
3431, 33syl 17 . . . . . . . 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 5689 . . . . . . . . . 10  |-  ( F  Fn  X  ->  dom  F  =  X )
3621, 28, 353syl 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 ) ) ) )  ->  dom  F  =  X )
37 toponuni 19928 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3821, 37syl 17 . . . . . . . . 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 2463 . . . . . . . 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 3501 . . . . . . 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 6009 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
dom  F )  -> 
( ( F "
w )  C_  m  <->  w 
C_  ( `' F " m ) ) )
4230, 40, 41syl2anc 665 . . . . . 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 213 . . . . 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 20728 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
45 topontop 19927 . . . . . . . . . 10  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
4621, 44, 453syl 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 ) ) ) )  ->  (KQ `  J
)  e.  Top )
47 elssuni 4245 . . . . . . . . . 10  |-  ( m  e.  (KQ `  J
)  ->  m  C_  U. (KQ `  J ) )
4847ad2antrl 732 . . . . . . . . 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 2422 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
5049clscld 20048 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  (
( cls `  (KQ `  J ) ) `  m )  e.  (
Clsd `  (KQ `  J
) ) )
5146, 48, 50syl2anc 665 . . . . . . . 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 20270 . . . . . . . 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 665 . . . . . . 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 20057 . . . . . . . . 9  |-  ( ( (KQ `  J )  e.  Top  /\  m  C_ 
U. (KQ `  J
) )  ->  m  C_  ( ( cls `  (KQ `  J ) ) `  m ) )
5546, 48, 54syl2anc 665 . . . . . . . 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 5219 . . . . . . . 8  |-  ( m 
C_  ( ( cls `  (KQ `  J ) ) `  m )  ->  ( `' F " m )  C_  ( `' F " ( ( cls `  (KQ `  J ) ) `  m ) ) )
5755, 56syl 17 . . . . . . 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 20074 . . . . . . 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 665 . . . . . 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 773 . . . . . . . 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 5219 . . . . . . . 8  |-  ( ( ( cls `  (KQ `  J ) ) `  m )  C_  ( F " z )  -> 
( `' F "
( ( cls `  (KQ `  J ) ) `  m ) )  C_  ( `' F " ( F
" z ) ) )
6260, 61syl 17 . . . . . . 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 466 . . . . . . . 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 20732 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  J )  ->  ( `' F " ( F
" z ) )  =  z )
6521, 63, 64syl2anc 665 . . . . . . 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 3500 . . . . . 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 3474 . . . . 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 3486 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
w  C_  u  <->  w  C_  ( `' F " m ) ) )
69 fveq2 5877 . . . . . . . 8  |-  ( u  =  ( `' F " m )  ->  (
( cls `  J
) `  u )  =  ( ( cls `  J ) `  ( `' F " m ) ) )
7069sseq1d 3491 . . . . . . 7  |-  ( u  =  ( `' F " m )  ->  (
( ( cls `  J
) `  u )  C_  z  <->  ( ( cls `  J ) `  ( `' F " m ) )  C_  z )
)
7168, 70anbi12d 715 . . . . . 6  |-  ( u  =  ( `' F " m )  ->  (
( w  C_  u  /\  ( ( cls `  J
) `  u )  C_  z )  <->  ( w  C_  ( `' F "
m )  /\  (
( cls `  J
) `  ( `' F " m ) ) 
C_  z ) ) )
7271rspcev 3182 . . . . 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 1262 . . . 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 2921 . . 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 2846 . 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 20337 . 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 668 1  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Nrm )  ->  J  e.  Nrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   {crab 2779    i^i cin 3435    C_ wss 3436   ~Pcpw 3979   U.cuni 4216    |-> cmpt 4479   `'ccnv 4848   dom cdm 4849   ran crn 4850   "cima 4852   Fun wfun 5591    Fn wfn 5592   ` cfv 5597  (class class class)co 6301   Topctop 19903  TopOnctopon 19904   Clsdccld 20017   clsccl 20019    Cn ccn 20226   Nrmcnrm 20312  KQckq 20694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-qtop 15393  df-top 19907  df-topon 19909  df-cld 20020  df-cls 20022  df-cn 20229  df-nrm 20319  df-kq 20695
This theorem is referenced by:  kqnrm  20753
  Copyright terms: Public domain W3C validator