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Theorem submgmrcl 18628
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 18626 . . 3 SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
21dmmptss 6234 . 2 dom SubMgm ⊆ Mgm
3 elfvdm 6922 . 2 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
42, 3sselid 3975 1 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3055  {crab 3426  𝒫 cpw 4597  dom cdm 5669  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Mgmcmgm 18571  SubMgmcsubmgm 18624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fv 6545  df-submgm 18626
This theorem is referenced by:  submgmss  18638  submgmcl  18640  submgmmgm  18641  subsubmgm  18643  resmgmhm2  18645
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