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Theorem submgmrcl 45336
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 45334 . . 3 SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
21dmmptss 6144 . 2 dom SubMgm ⊆ Mgm
3 elfvdm 6806 . 2 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
42, 3sselid 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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