| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 18757 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
| 2 | 1 | dmmptss 6241 | . 2 ⊢ dom SubMgm ⊆ Mgm |
| 3 | elfvdm 6915 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
| 4 | 2, 3 | sselid 3934 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∈ wcel 2142 ∀wral 3078 {crab 3415 𝒫 cpw 4561 dom cdm 5660 ‘cfv 6536 (class class class)co 7412 Basecbs 17275 +gcplusg 17316 Mgmcmgm 18702 SubMgmcsubmgm 18755 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-xp 5666 df-rel 5667 df-cnv 5668 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fv 6544 df-submgm 18757 |
| This theorem is used by: submgmss 18769 submgmcl 18771 submgmmgm 18772 subsubmgm 18774 resmgmhm2 18776 |
| Copyright terms: Public domain | W3C validator |