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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 16890. (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 16890 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝑁‘𝑈)) |
5 | 1, 2, 4 | syl2anc 586 | 1 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 Moorecmre 16855 mrClscmrc 16856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-mre 16859 df-mrc 16860 |
This theorem is referenced by: submrc 16901 mrieqvlemd 16902 mrieqv2d 16912 mreexmrid 16916 mreexexlem2d 16918 mreexexlem3d 16919 mreexfidimd 16923 isacs2 16926 acsmap2d 17791 cycsubg2cl 18356 odf1o1 18699 gsumzsplit 19049 gsumzoppg 19066 gsumpt 19084 dprdfeq0 19146 dprdspan 19151 subgdmdprd 19158 subgdprd 19159 dprd2dlem1 19165 dprd2da 19166 dmdprdsplit2lem 19169 pgpfac1lem1 19198 pgpfac1lem3a 19200 pgpfac1lem3 19201 pgpfac1lem5 19203 pgpfaclem2 19206 proot1mul 39806 |
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