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Mirrors > Home > MPE Home > Th. List > mrcssvd | Structured version Visualization version GIF version |
Description: The Moore closure of a set is a subset of the base. Deduction form of mrcssv 16884. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrcssd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrcssd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
Ref | Expression |
---|---|
mrcssvd | ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcssd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mrcssd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | 2 | mrcssv 16884 | . 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑁‘𝐵) ⊆ 𝑋) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → (𝑁‘𝐵) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6354 Moorecmre 16852 mrClscmrc 16853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-int 4876 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-mre 16856 df-mrc 16857 |
This theorem is referenced by: mressmrcd 16897 mreexexlem2d 16915 mreacs 16928 acsmap2d 17788 gsumwspan 18010 cntzspan 18963 dprd2dlem1 19162 pgpfaclem2 19203 ismrcd2 39294 |
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