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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 16880. (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 16880 | . 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 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 Moorecmre 16845 mrClscmrc 16846 |
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 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-mre 16849 df-mrc 16850 |
This theorem is referenced by: submrc 16891 mrieqvlemd 16892 mrieqv2d 16902 mreexmrid 16906 mreexexlem2d 16908 mreexexlem3d 16909 mreexfidimd 16913 isacs2 16916 acsmap2d 17781 cycsubg2cl 18346 odf1o1 18689 gsumzsplit 19040 gsumzoppg 19057 gsumpt 19075 dprdfeq0 19137 dprdspan 19142 subgdmdprd 19149 subgdprd 19150 dprd2dlem1 19156 dprd2da 19157 dmdprdsplit2lem 19160 pgpfac1lem1 19189 pgpfac1lem3a 19191 pgpfac1lem3 19192 pgpfac1lem5 19194 pgpfaclem2 19197 proot1mul 40143 |
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