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Theorem mrcsncl 13514
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 3759 . 2  |-  ( U  e.  X  ->  { U }  C_  X )
2 mrcfval.f . . 3  |-  F  =  (mrCls `  C )
32mrccl 13513 . 2  |-  ( ( C  e.  (Moore `  X )  /\  { U }  C_  X )  ->  ( F `  { U } )  e.  C )
41, 3sylan2 460 1  |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  pgpfac1lem1  15309  pgpfac1lem2  15310  pgpfac1lem3a  15311  pgpfac1lem3  15312  pgpfac1lem4  15313  pgpfac1lem5  15314  pgpfaclem1  15316  pgpfaclem2  15317
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-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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