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Theorem mrcsncl 16954
 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 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)

Proof of Theorem mrcsncl
StepHypRef Expression
1 snssi 4701 . 2 (𝑈𝑋 → {𝑈} ⊆ 𝑋)
2 mrcfval.f . . 3 𝐹 = (mrCls‘𝐶)
32mrccl 16953 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈} ⊆ 𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
41, 3sylan2 595 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  {csn 4525  ‘cfv 6340  Moorecmre 16924  mrClscmrc 16925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-mre 16928  df-mrc 16929 This theorem is referenced by:  pgpfac1lem1  19277  pgpfac1lem2  19278  pgpfac1lem3a  19279  pgpfac1lem3  19280  pgpfac1lem4  19281  pgpfac1lem5  19282  pgpfaclem1  19284  pgpfaclem2  19285
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