Theorem submgmrcl 44056

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 44054	. . 3 ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡})
2	1	dmmptss 6097	. 2 ⊢ dom SubMgm ⊆ Mgm
3		elfvdm 6704	. 2 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ dom SubMgm)
4	2, 3	sseldi 3967	1 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3140  {crab 3144  𝒫 cpw 4541  dom cdm 5557  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  Mgmcmgm 17852  SubMgmcsubmgm 44052

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fv 6365  df-submgm 44054

This theorem is referenced by:  submgmss  44066  submgmcl  44068  submgmmgm  44069  subsubmgm  44071  resmgmhm2  44073