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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 18624 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
2 | 1 | dmmptss 6240 | . 2 ⊢ dom SubMgm ⊆ Mgm |
3 | elfvdm 6928 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
4 | 2, 3 | sselid 3980 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∀wral 3060 {crab 3431 𝒫 cpw 4602 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Mgmcmgm 18569 SubMgmcsubmgm 18622 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fv 6551 df-submgm 18624 |
This theorem is referenced by: submgmss 18636 submgmcl 18638 submgmmgm 18639 subsubmgm 18641 resmgmhm2 18643 |
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