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Theorem submrcl 17393
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 17383 . . 3 SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
21dmmptss 5669 . 2 dom SubMnd ⊆ Mnd
3 elfvdm 6258 . 2 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ dom SubMnd)
42, 3sseldi 3634 1 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  {crab 2945  𝒫 cpw 4191  dom cdm 5143  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341  SubMndcsubmnd 17381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-submnd 17383
This theorem is referenced by:  submss  17397  subm0cl  17399  submcl  17400  submmnd  17401  subm0  17403  subsubm  17404  resmhm2  17407  gsumsubm  17420  gsumwsubmcl  17422  submmulgcl  17632  oppgsubm  17838  lsmub1x  18107  lsmub2x  18108  lsmsubm  18114  submarchi  29868
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