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Theorem mhmrcl1 17546
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 17543 . 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 7028 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502   MndHom cmhm 17541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-dm 5260  df-iota 5993  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-mhm 17543
This theorem is referenced by:  mhmf1o  17553  resmhm2  17568  resmhm2b  17569  mhmco  17570  mhmeql  17572  pwsco2mhm  17579  gsumwmhm  17590  mhmmulg  17791  mhmvlin  20420  mhmhmeotmd  30313
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