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Theorem isnrm 19602
Description: The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
isnrm  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
Distinct variable group:    x, y, z, J

Proof of Theorem isnrm
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5864 . . . . 5  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
21ineq1d 3699 . . . 4  |-  ( j  =  J  ->  (
( Clsd `  j )  i^i  ~P x )  =  ( ( Clsd `  J
)  i^i  ~P x
) )
3 fveq2 5864 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5866 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
54sseq1d 3531 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
65anbi2d 703 . . . . 5  |-  ( j  =  J  ->  (
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
76rexeqbi1dv 3067 . . . 4  |-  ( j  =  J  ->  ( E. z  e.  j 
( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `
 z )  C_  x ) ) )
82, 7raleqbidv 3072 . . 3  |-  ( j  =  J  ->  ( A. y  e.  (
( Clsd `  j )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
98raleqbi1dv 3066 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  (
( Clsd `  j )  i^i  ~P x ) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j
) `  z )  C_  x )  <->  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
10 df-nrm 19584 . 2  |-  Nrm  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  ( ( Clsd `  j
)  i^i  ~P x
) E. z  e.  j  ( y  C_  z  /\  ( ( cls `  j ) `  z
)  C_  x ) }
119, 10elrab2 3263 1  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  ( ( Clsd `  J
)  i^i  ~P x
) E. z  e.  J  ( y  C_  z  /\  ( ( cls `  J ) `  z
)  C_  x )
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   ` cfv 5586   Topctop 19161   Clsdccld 19283   clsccl 19285   Nrmcnrm 19577
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-nrm 19584
This theorem is referenced by:  nrmtop  19603  nrmsep3  19622  isnrm2  19625  kqnrmlem1  19979  kqnrmlem2  19980  nrmhmph  20030
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