Theorem submgmrcl 18729

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 18727	. . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
2	1	dmmptss 6228	. 2 ⊢ dom SubMgm ⊆ Mgm
3		elfvdm 6901	. 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
4	2, 3	sselid 3934	1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2142  ∀wral 3076  {crab 3414  𝒫 cpw 4555  dom cdm 5647  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  Mgmcmgm 18672  SubMgmcsubmgm 18725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fv 6529  df-submgm 18727

This theorem is referenced by:  submgmss  18739  submgmcl  18741  submgmmgm  18742  subsubmgm  18744  resmgmhm2  18746