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| Description: A set is contained in its Moore closure. Deduction form of mrcssid 17660. (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 17660 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝑁‘𝑈)) | 
| 5 | 1, 2, 4 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Moorecmre 17625 mrClscmrc 17626 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-mre 17629 df-mrc 17630 | 
| This theorem is referenced by: submrc 17671 mrieqvlemd 17672 mrieqv2d 17682 mreexmrid 17686 mreexexlem2d 17688 mreexexlem3d 17689 mreexfidimd 17693 isacs2 17696 acsmap2d 18600 cycsubg2cl 19229 odf1o1 19590 gsumzsplit 19945 gsumzoppg 19962 gsumpt 19980 dprdfeq0 20042 dprdspan 20047 subgdmdprd 20054 subgdprd 20055 dprd2dlem1 20061 dprd2da 20062 dmdprdsplit2lem 20065 pgpfac1lem1 20094 pgpfac1lem3a 20096 pgpfac1lem3 20097 pgpfac1lem5 20099 pgpfaclem2 20102 proot1mul 43206 | 
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