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Theorem mgmhmrcl 46161
Description: Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
Assertion
Ref Expression
mgmhmrcl (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))

Proof of Theorem mgmhmrcl
Dummy variables 𝑡 𝑠 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mgmhm 46159 . 2 MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦))})
21elmpocl 7596 1 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  {crab 3406  cfv 6497  (class class class)co 7358  m cmap 8768  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500   MgmHom cmgmhm 46157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgmhm 46159
This theorem is referenced by:  ismgmhm  46163  mgmhmf1o  46167  resmgmhm  46178  resmgmhm2  46179  resmgmhm2b  46180  mgmhmco  46181  mgmhmima  46182  mgmhmeql  46183
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