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Theorem mrieqvlemd 14900
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 14909 and mrieqv2d 14910. 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 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  A  e.  (Moore `  X
) )
3 mrieqvlemd.2 . . . 4  |-  N  =  (mrCls `  A )
4 undif1 3908 . . . . . 6  |-  ( ( S  \  { Y } )  u.  { Y } )  =  ( S  u.  { Y } )
5 mrieqvlemd.3 . . . . . . . . . 10  |-  ( ph  ->  S  C_  X )
65adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  X )
76ssdifssd 3647 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  X
)
82, 3, 7mrcssidd 14896 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
9 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  Y  e.  ( N `  ( S  \  { Y } ) ) )
109snssd 4178 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  { Y }  C_  ( N `  ( S  \  { Y } ) ) )
118, 10unssd 3685 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( ( S  \  { Y } )  u. 
{ Y } ) 
C_  ( N `  ( S  \  { Y } ) ) )
124, 11syl5eqssr 3554 . . . . 5  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  u.  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
1312unssad 3686 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  ( N `  ( S  \  { Y } ) ) )
14 difssd 3637 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  S
)
152, 3, 13, 14mressmrcd 14898 . . 3  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  S
)  =  ( N `
 ( S  \  { Y } ) ) )
1615eqcomd 2475 . 2  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  ( S  \  { Y }
) )  =  ( N `  S ) )
171, 3, 5mrcssidd 14896 . . . . 5  |-  ( ph  ->  S  C_  ( N `  S ) )
18 mrieqvlemd.4 . . . . 5  |-  ( ph  ->  Y  e.  S )
1917, 18sseldd 3510 . . . 4  |-  ( ph  ->  Y  e.  ( N `
 S ) )
2019adantr 465 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  S
) )
21 simpr 461 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  ( N `  ( S  \  { Y } ) )  =  ( N `
 S ) )
2220, 21eleqtrrd 2558 . 2  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  ( S  \  { Y }
) ) )
2316, 22impbida 830 1  |-  ( ph  ->  ( Y  e.  ( N `  ( S 
\  { Y }
) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3478    u. cun 3479    C_ wss 3481   {csn 4033   ` cfv 5594  Moorecmre 14853  mrClscmrc 14854
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 14857  df-mrc 14858
This theorem is referenced by:  mrieqvd  14909  mrieqv2d  14910
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