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Theorem submgmrcl 42779
Description: Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
Assertion
Ref Expression
submgmrcl (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)

Proof of Theorem submgmrcl
Dummy variables 𝑡 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-submgm 42777 . . 3 SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
21dmmptss 5885 . 2 dom SubMgm ⊆ Mgm
3 elfvdm 6478 . 2 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
42, 3sseldi 3818 1 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3089  {crab 3093  𝒫 cpw 4378  dom cdm 5355  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  Mgmcmgm 17626  SubMgmcsubmgm 42775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fv 6143  df-submgm 42777
This theorem is referenced by:  submgmss  42789  submgmcl  42791  submgmmgm  42792  subsubmgm  42794  resmgmhm2  42796
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