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Theorem submrcl 17957
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 17947 . 2 SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
21mptrcl 6772 1 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3142  {crab 3146  𝒫 cpw 4541  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  0gc0g 16705  Mndcmnd 17902  SubMndcsubmnd 17945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-rel 5560  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fv 6359  df-submnd 17947
This theorem is referenced by:  submss  17961  subm0cl  17963  submcl  17964  submmnd  17965  subm0  17967  subsubm  17968  resmhm2  17971  gsumsubm  17984  gsumwsubmcl  17986  submmulgcl  18202  oppgsubm  18422  lsmub1x  18693  lsmub2x  18694  lsmsubm  18700  submarchi  30730
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