MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcsncl Structured version   Visualization version   GIF version

Theorem mrcsncl 16212
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 4315 . 2 (𝑈𝑋 → {𝑈} ⊆ 𝑋)
2 mrcfval.f . . 3 𝐹 = (mrCls‘𝐶)
32mrccl 16211 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈} ⊆ 𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
41, 3sylan2 491 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘{𝑈}) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560  {csn 4155  cfv 5857  Moorecmre 16182  mrClscmrc 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-mre 16186  df-mrc 16187
This theorem is referenced by:  pgpfac1lem1  18413  pgpfac1lem2  18414  pgpfac1lem3a  18415  pgpfac1lem3  18416  pgpfac1lem4  18417  pgpfac1lem5  18418  pgpfaclem1  18420  pgpfaclem2  18421
  Copyright terms: Public domain W3C validator