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Theorem mrieqvlemd 14579
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 14588 and mrieqv2d 14589. 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 3766 . . . . . 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 3506 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  X
)
82, 3, 7mrcssidd 14575 . . . . . . 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 4030 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  { Y }  C_  ( N `  ( S  \  { Y } ) ) )
118, 10unssd 3544 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( ( S  \  { Y } )  u. 
{ Y } ) 
C_  ( N `  ( S  \  { Y } ) ) )
124, 11syl5eqssr 3413 . . . . 5  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  u.  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
1312unssad 3545 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  ( N `  ( S  \  { Y } ) ) )
14 difssd 3496 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  S
)
152, 3, 13, 14mressmrcd 14577 . . 3  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  S
)  =  ( N `
 ( S  \  { Y } ) ) )
1615eqcomd 2448 . 2  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  ( S  \  { Y }
) )  =  ( N `  S ) )
171, 3, 5mrcssidd 14575 . . . . 5  |-  ( ph  ->  S  C_  ( N `  S ) )
18 mrieqvlemd.4 . . . . 5  |-  ( ph  ->  Y  e.  S )
1917, 18sseldd 3369 . . . 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 2520 . 2  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  ( S  \  { Y }
) ) )
2316, 22impbida 828 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 1369    e. wcel 1756    \ cdif 3337    u. cun 3338    C_ wss 3340   {csn 3889   ` cfv 5430  Moorecmre 14532  mrClscmrc 14533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-mre 14536  df-mrc 14537
This theorem is referenced by:  mrieqvd  14588  mrieqv2d  14589
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