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Theorem mhmrcl2 17560
Description: Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
Assertion
Ref Expression
mhmrcl2 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)

Proof of Theorem mhmrcl2
Dummy variables 𝑓 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mhm 17556 . 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))})
21elmpt2cl2 7044 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Mndcmnd 17515   MndHom cmhm 17554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-dm 5276  df-iota 6012  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-mhm 17556
This theorem is referenced by:  mhmf1o  17566  resmhm  17580  mhmco  17583  mhmima  17584  pwsco2mhm  17592  gsumwmhm  17603  mhmmulg  17804  mhmhmeotmd  30303
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