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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 45334 | . . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | |
2 | 1 | dmmptss 6144 | . 2 ⊢ dom SubMgm ⊆ Mgm |
3 | elfvdm 6806 | . 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm) | |
4 | 2, 3 | sselid 3919 | 1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3064 {crab 3068 𝒫 cpw 4533 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 Mgmcmgm 18324 SubMgmcsubmgm 45332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fv 6441 df-submgm 45334 |
This theorem is referenced by: submgmss 45346 submgmcl 45348 submgmmgm 45349 subsubmgm 45351 resmgmhm2 45353 |
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