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Mirrors > Home > MPE Home > Th. List > Mathboxes > submgmrcl | Structured version Visualization version GIF version |
Description: Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
Ref | Expression |
---|---|
submgmrcl | ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-submgm 42777 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
2 | 1 | dmmptss 5885 | . 2 ⊢ dom SubMgm ⊆ Mgm |
3 | elfvdm 6478 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
4 | 2, 3 | sseldi 3818 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3089 {crab 3093 𝒫 cpw 4378 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Mgmcmgm 17626 SubMgmcsubmgm 42775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fv 6143 df-submgm 42777 |
This theorem is referenced by: submgmss 42789 submgmcl 42791 submgmmgm 42792 subsubmgm 42794 resmgmhm2 42796 |
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