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Theorem mrieqvlemd 13781
Description: In a Moore system, if  Y is a member of  S,  ( S  \  { Y } ) and  S have the same closure if and only if  Y is in the closure of  ( S  \  { Y } ). Used in the proof of mrieqvd 13790 and mrieqv2d 13791. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrieqvlemd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mrieqvlemd.2  |-  N  =  (mrCls `  A )
mrieqvlemd.3  |-  ( ph  ->  S  C_  X )
mrieqvlemd.4  |-  ( ph  ->  Y  e.  S )
Assertion
Ref Expression
mrieqvlemd  |-  ( ph  ->  ( Y  e.  ( N `  ( S 
\  { Y }
) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) ) )

Proof of Theorem mrieqvlemd
StepHypRef Expression
1 mrieqvlemd.1 . . . . 5  |-  ( ph  ->  A  e.  (Moore `  X ) )
21adantr 452 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  A  e.  (Moore `  X
) )
3 mrieqvlemd.2 . . . 4  |-  N  =  (mrCls `  A )
4 undif1 3646 . . . . . 6  |-  ( ( S  \  { Y } )  u.  { Y } )  =  ( S  u.  { Y } )
5 mrieqvlemd.3 . . . . . . . . . 10  |-  ( ph  ->  S  C_  X )
65adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  X )
76ssdifssd 3428 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  X
)
82, 3, 7mrcssidd 13777 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
9 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  Y  e.  ( N `  ( S  \  { Y } ) ) )
109snssd 3886 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  { Y }  C_  ( N `  ( S  \  { Y } ) ) )
118, 10unssd 3466 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( ( S  \  { Y } )  u. 
{ Y } ) 
C_  ( N `  ( S  \  { Y } ) ) )
124, 11syl5eqssr 3336 . . . . 5  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  u.  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
1312unssad 3467 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  ( N `  ( S  \  { Y } ) ) )
14 difssd 3418 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  S
)
152, 3, 13, 14mressmrcd 13779 . . 3  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  S
)  =  ( N `
 ( S  \  { Y } ) ) )
1615eqcomd 2392 . 2  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  ( S  \  { Y }
) )  =  ( N `  S ) )
171, 3, 5mrcssidd 13777 . . . . 5  |-  ( ph  ->  S  C_  ( N `  S ) )
18 mrieqvlemd.4 . . . . 5  |-  ( ph  ->  Y  e.  S )
1917, 18sseldd 3292 . . . 4  |-  ( ph  ->  Y  e.  ( N `
 S ) )
2019adantr 452 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  S
) )
21 simpr 448 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  ( N `  ( S  \  { Y } ) )  =  ( N `
 S ) )
2220, 21eleqtrrd 2464 . 2  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  ( S  \  { Y }
) ) )
2316, 22impbida 806 1  |-  ( ph  ->  ( Y  e.  ( N `  ( S 
\  { Y }
) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3260    u. cun 3261    C_ wss 3263   {csn 3757   ` cfv 5394  Moorecmre 13734  mrClscmrc 13735
This theorem is referenced by:  mrieqvd  13790  mrieqv2d  13791
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-mre 13738  df-mrc 13739
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