![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16588. (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 16588 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝑁‘𝑈)) |
5 | 1, 2, 4 | syl2anc 580 | 1 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ⊆ wss 3767 ‘cfv 6099 Moorecmre 16553 mrClscmrc 16554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-int 4666 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-fv 6107 df-mre 16557 df-mrc 16558 |
This theorem is referenced by: submrc 16599 mrieqvlemd 16600 mrieqv2d 16610 mreexmrid 16614 mreexexlem2d 16616 mreexexlem3d 16617 mreexfidimd 16621 isacs2 16624 acsmap2d 17490 cycsubg2cl 17941 odf1o1 18296 gsumzsplit 18638 gsumzoppg 18655 gsumpt 18672 dprdfeq0 18733 dprdspan 18738 subgdmdprd 18745 subgdprd 18746 dprd2dlem1 18752 dprd2da 18753 dmdprdsplit2lem 18756 pgpfac1lem1 18785 pgpfac1lem3a 18787 pgpfac1lem3 18788 pgpfac1lem5 18790 pgpfaclem2 18793 proot1mul 38549 |
Copyright terms: Public domain | W3C validator |