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Theorem mhmrcl1 17691
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)

Proof of Theorem mhmrcl1
Dummy variables 𝑓 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 17688 . 2 MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ∧ (𝑓‘(0g𝑠)) = (0g𝑡))})
21elmpt2cl1 7137 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  Basecbs 16222  +gcplusg 16305  0gc0g 16453  Mndcmnd 17647   MndHom cmhm 17686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-dm 5352  df-iota 6086  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-mhm 17688
This theorem is referenced by:  mhmf1o  17698  resmhm2  17713  resmhm2b  17714  mhmco  17715  mhmeql  17717  pwsco2mhm  17724  gsumwmhm  17736  mhmmulg  17934  mhmvlin  20570  mhmhmeotmd  30507
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