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Theorem mrieqvlemd 13547
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 13556 and mrieqv2d 13557. 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 3542 . . . . . 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 3327 . . . . . . . 8  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  X
)
82, 3, 7mrcssidd 13543 . . . . . . 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 3776 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  { Y }  C_  ( N `  ( S  \  { Y } ) ) )
118, 10unssd 3364 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( ( S  \  { Y } )  u. 
{ Y } ) 
C_  ( N `  ( S  \  { Y } ) ) )
124, 11syl5eqssr 3236 . . . . 5  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  u.  { Y } )  C_  ( N `  ( S  \  { Y } ) ) )
1312unssad 3365 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  ->  S  C_  ( N `  ( S  \  { Y } ) ) )
14 difssd 3317 . . . 4  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( S  \  { Y } )  C_  S
)
152, 3, 13, 14mressmrcd 13545 . . 3  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  S
)  =  ( N `
 ( S  \  { Y } ) ) )
1615eqcomd 2301 . 2  |-  ( (
ph  /\  Y  e.  ( N `  ( S 
\  { Y }
) ) )  -> 
( N `  ( S  \  { Y }
) )  =  ( N `  S ) )
171, 3, 5mrcssidd 13543 . . . . 5  |-  ( ph  ->  S  C_  ( N `  S ) )
18 mrieqvlemd.4 . . . . 5  |-  ( ph  ->  Y  e.  S )
1917, 18sseldd 3194 . . . 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 2373 . 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 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrieqvd  13556  mrieqv2d  13557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
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