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Theorem submgmrcl 18626
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 18624 . . 3 SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
21dmmptss 6240 . 2 dom SubMgm ⊆ Mgm
3 elfvdm 6928 . 2 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
42, 3sselid 3980 1 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3060  {crab 3431  𝒫 cpw 4602  dom cdm 5676  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Mgmcmgm 18569  SubMgmcsubmgm 18622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-submgm 18624
This theorem is referenced by:  submgmss  18636  submgmcl  18638  submgmmgm  18639  subsubmgm  18641  resmgmhm2  18643
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