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Theorem mgmhmrcl 43880
 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 43878 . 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 7377 1 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  {crab 3147  ‘cfv 6352  (class class class)co 7148   ↑m cmap 8396  Basecbs 16473  +gcplusg 16555  Mgmcmgm 17840   MgmHom cmgmhm 43876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-dm 5564  df-iota 6312  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-mgmhm 43878 This theorem is referenced by:  ismgmhm  43882  mgmhmf1o  43886  resmgmhm  43897  resmgmhm2  43898  resmgmhm2b  43899  mgmhmco  43900  mgmhmima  43901  mgmhmeql  43902
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