![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 44400 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
2 | 1 | dmmptss 6062 | . 2 ⊢ dom SubMgm ⊆ Mgm |
3 | elfvdm 6677 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
4 | 2, 3 | sseldi 3913 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∀wral 3106 {crab 3110 𝒫 cpw 4497 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Mgmcmgm 17842 SubMgmcsubmgm 44398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 df-submgm 44400 |
This theorem is referenced by: submgmss 44412 submgmcl 44414 submgmmgm 44415 subsubmgm 44417 resmgmhm2 44419 |
Copyright terms: Public domain | W3C validator |