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Mirrors > Home > MPE Home > Th. List > mrcssidd | Structured version Visualization version GIF version |
Description: A set is contained in its Moore closure. Deduction form of mrcssid 17004. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrcssidd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrcssidd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrcssidd.3 | ⊢ (𝜑 → 𝑈 ⊆ 𝑋) |
Ref | Expression |
---|---|
mrcssidd | ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcssidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mrcssidd.3 | . 2 ⊢ (𝜑 → 𝑈 ⊆ 𝑋) | |
3 | mrcssidd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
4 | 3 | mrcssid 17004 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝑁‘𝑈)) |
5 | 1, 2, 4 | syl2anc 587 | 1 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 ‘cfv 6350 Moorecmre 16969 mrClscmrc 16970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-int 4847 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-mre 16973 df-mrc 16974 |
This theorem is referenced by: submrc 17015 mrieqvlemd 17016 mrieqv2d 17026 mreexmrid 17030 mreexexlem2d 17032 mreexexlem3d 17033 mreexfidimd 17037 isacs2 17040 acsmap2d 17918 cycsubg2cl 18485 odf1o1 18828 gsumzsplit 19179 gsumzoppg 19196 gsumpt 19214 dprdfeq0 19276 dprdspan 19281 subgdmdprd 19288 subgdprd 19289 dprd2dlem1 19295 dprd2da 19296 dmdprdsplit2lem 19299 pgpfac1lem1 19328 pgpfac1lem3a 19330 pgpfac1lem3 19331 pgpfac1lem5 19333 pgpfaclem2 19336 proot1mul 40637 |
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