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Theorem submrcl 17660
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 17650 . . 3 SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ((0g𝑠) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡)})
21dmmptss 5851 . 2 dom SubMnd ⊆ Mnd
3 elfvdm 6444 . 2 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ dom SubMnd)
42, 3sseldi 3797 1 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wral 3090  {crab 3094  𝒫 cpw 4350  dom cdm 5313  cfv 6102  (class class class)co 6879  Basecbs 16183  +gcplusg 16266  0gc0g 16414  Mndcmnd 17608  SubMndcsubmnd 17648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-xp 5319  df-rel 5320  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fv 6110  df-submnd 17650
This theorem is referenced by:  submss  17664  subm0cl  17666  submcl  17667  submmnd  17668  subm0  17670  subsubm  17671  resmhm2  17674  gsumsubm  17687  gsumwsubmcl  17689  submmulgcl  17897  oppgsubm  18103  lsmub1x  18373  lsmub2x  18374  lsmsubm  18380  submarchi  30255
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