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Theorem mrcsncl 17499
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 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcsncl ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ ∈ 𝑋) β†’ (πΉβ€˜{π‘ˆ}) ∈ 𝐢)

Proof of Theorem mrcsncl
StepHypRef Expression
1 snssi 4773 . 2 (π‘ˆ ∈ 𝑋 β†’ {π‘ˆ} βŠ† 𝑋)
2 mrcfval.f . . 3 𝐹 = (mrClsβ€˜πΆ)
32mrccl 17498 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ {π‘ˆ} βŠ† 𝑋) β†’ (πΉβ€˜{π‘ˆ}) ∈ 𝐢)
41, 3sylan2 594 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ ∈ 𝑋) β†’ (πΉβ€˜{π‘ˆ}) ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3915  {csn 4591  β€˜cfv 6501  Moorecmre 17469  mrClscmrc 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-mre 17473  df-mrc 17474
This theorem is referenced by:  pgpfac1lem1  19860  pgpfac1lem2  19861  pgpfac1lem3a  19862  pgpfac1lem3  19863  pgpfac1lem4  19864  pgpfac1lem5  19865  pgpfaclem1  19867  pgpfaclem2  19868
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