Theorem submgmrcl 45224

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.

Step	Hyp	Ref	Expression
1		df-submgm 45222	. . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
2	1	dmmptss 6133	. 2 ⊢ dom SubMgm ⊆ Mgm
3		elfvdm 6788	. 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
4	2, 3	sselid 3915	1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3063  {crab 3067  𝒫 cpw 4530  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mgmcmgm 18239  SubMgmcsubmgm 45220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426  df-submgm 45222

This theorem is referenced by:  submgmss  45234  submgmcl  45236  submgmmgm  45237  subsubmgm  45239  resmgmhm2  45241