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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 17662. (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 17662 | . 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 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 Moorecmre 17627 mrClscmrc 17628 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-mre 17631 df-mrc 17632 |
This theorem is referenced by: submrc 17673 mrieqvlemd 17674 mrieqv2d 17684 mreexmrid 17688 mreexexlem2d 17690 mreexexlem3d 17691 mreexfidimd 17695 isacs2 17698 acsmap2d 18613 cycsubg2cl 19242 odf1o1 19605 gsumzsplit 19960 gsumzoppg 19977 gsumpt 19995 dprdfeq0 20057 dprdspan 20062 subgdmdprd 20069 subgdprd 20070 dprd2dlem1 20076 dprd2da 20077 dmdprdsplit2lem 20080 pgpfac1lem1 20109 pgpfac1lem3a 20111 pgpfac1lem3 20112 pgpfac1lem5 20114 pgpfaclem2 20117 proot1mul 43183 |
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