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Theorem mrieqvlemd 13531
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 13540 and mrieqv2d 13541. 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 451 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  A  e.  (Moore `  X
) )
3 mrieqvlemd.2 . . . 4  |-  N  =  (mrCls `  A )
4 undif1 3529 . . . . . 6  |-  ( ( S  \  { Y } )  u.  { Y } )  =  ( S  u.  { Y } )
5 mrieqvlemd.3 . . . . . . . . . 10  |-  ( ph  ->  S  C_  X )
65adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  X )
76ssdifssd 3314 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  X
)
82, 3, 7mrcssidd 13527 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
9 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  Y  e.  ( N `  ( S  \  { Y } ) ) )
109snssd 3760 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  { Y }  C_  ( N `  ( S  \  { Y } ) ) )
118, 10unssd 3351 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( ( S  \  { Y } )  u. 
{ Y } ) 
C_  ( N `  ( S  \  { Y } ) ) )
124, 11syl5eqssr 3223 . . . . 5  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  u.  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
1312unssad 3352 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  ( N `  ( S  \  { Y } ) ) )
14 difssd 3304 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  S
)
152, 3, 13, 14mressmrcd 13529 . . 3  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  S
)  =  ( N `
 ( S  \  { Y } ) ) )
1615eqcomd 2288 . 2  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  ( S  \  { Y }
) )  =  ( N `  S ) )
171, 3, 5mrcssidd 13527 . . . . 5  |-  ( ph  ->  S  C_  ( N `  S ) )
18 mrieqvlemd.4 . . . . 5  |-  ( ph  ->  Y  e.  S )
1917, 18sseldd 3181 . . . 4  |-  ( ph  ->  Y  e.  ( N `
 S ) )
2019adantr 451 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  S
) )
21 simpr 447 . . 3  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  ( N `  ( S  \  { Y } ) )  =  ( N `
 S ) )
2220, 21eleqtrrd 2360 . 2  |-  ( (
ph  /\  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) )  ->  Y  e.  ( N `  ( S  \  { Y }
) ) )
2316, 22impbida 805 1  |-  ( ph  ->  ( Y  e.  ( N `  ( S 
\  { Y }
) )  <->  ( N `  ( S  \  { Y } ) )  =  ( N `  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  mrieqvd  13540  mrieqv2d  13541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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