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Theorem mrcsncl 13764
Description: The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcsncl  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C
)

Proof of Theorem mrcsncl
StepHypRef Expression
1 snssi 3885 . 2  |-  ( U  e.  X  ->  { U }  C_  X )
2 mrcfval.f . . 3  |-  F  =  (mrCls `  C )
32mrccl 13763 . 2  |-  ( ( C  e.  (Moore `  X )  /\  { U }  C_  X )  ->  ( F `  { U } )  e.  C )
41, 3sylan2 461 1  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   {csn 3757   ` cfv 5394  Moorecmre 13734  mrClscmrc 13735
This theorem is referenced by:  pgpfac1lem1  15559  pgpfac1lem2  15560  pgpfac1lem3a  15561  pgpfac1lem3  15562  pgpfac1lem4  15563  pgpfac1lem5  15564  pgpfaclem1  15566  pgpfaclem2  15567
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-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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