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Theorem submgmrcl 41553
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 41551 . . 3 SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
21dmmptss 5533 . 2 dom SubMgm ⊆ Mgm
3 elfvdm 6114 . 2 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
42, 3sseldi 3565 1 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  wral 2895  {crab 2899  𝒫 cpw 4107  dom cdm 5027  cfv 5789  (class class class)co 6526  Basecbs 15643  +gcplusg 15716  Mgmcmgm 17011  SubMgmcsubmgm 41549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fv 5797  df-submgm 41551
This theorem is referenced by:  submgmss  41563  submgmcl  41565  submgmmgm  41566  subsubmgm  41568  resmgmhm2  41570
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