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| Mirrors > Home > MPE Home > Th. List > 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 18706 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
| 2 | 1 | dmmptss 6261 | . 2 ⊢ dom SubMgm ⊆ Mgm |
| 3 | elfvdm 6943 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
| 4 | 2, 3 | sselid 3981 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 {crab 3436 𝒫 cpw 4600 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 SubMgmcsubmgm 18704 |
| 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-pr 5432 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-submgm 18706 |
| This theorem is referenced by: submgmss 18718 submgmcl 18720 submgmmgm 18721 subsubmgm 18723 resmgmhm2 18725 |
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