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Theorem mhmrcl2 17693
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 17689 . 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 7139 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3118  {crab 3122  cfv 6124  (class class class)co 6906  𝑚 cmap 8123  Basecbs 16223  +gcplusg 16306  0gc0g 16454  Mndcmnd 17648   MndHom cmhm 17687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-xp 5349  df-dm 5353  df-iota 6087  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-mhm 17689
This theorem is referenced by:  mhmf1o  17699  resmhm  17713  mhmco  17716  mhmima  17717  pwsco2mhm  17725  gsumwmhm  17737  mhmmulg  17935  mhmhmeotmd  30519
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