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Theorem submrcl 17169
Description: Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
submrcl (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)

Proof of Theorem submrcl
Dummy variables 𝑡 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-submnd 17159 . . 3 SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
21dmmptss 5548 . 2 dom SubMnd ⊆ Mnd
3 elfvdm 6130 . 2 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ dom SubMnd)
42, 3sseldi 3566 1 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  wral 2896  {crab 2900  𝒫 cpw 4108  dom cdm 5038  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117  SubMndcsubmnd 17157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-submnd 17159
This theorem is referenced by:  submss  17173  subm0cl  17175  submcl  17176  submmnd  17177  subm0  17179  subsubm  17180  resmhm2  17183  gsumsubm  17196  gsumwsubmcl  17198  submmulgcl  17408  oppgsubm  17615  lsmub1x  17884  lsmub2x  17885  lsmsubm  17891  submarchi  29071
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