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Theorem mhmrcl1 18715
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 18711 . 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))})
21elmpocl1 7645 1 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  {crab 3426  cfv 6536  (class class class)co 7404  m cmap 8819  Basecbs 17151  +gcplusg 17204  0gc0g 17392  Mndcmnd 18665   MndHom cmhm 18709
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-iota 6488  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-mhm 18711
This theorem is referenced by:  mhmf1o  18724  resmhm2  18744  resmhm2b  18745  mhmco  18746  mhmeql  18749  pwsco2mhm  18756  gsumwmhm  18768  mhmmulg  19040  mhmvlin  22250  mhmimasplusg  32700  mhmhmeotmd  33437  mhmcompl  41658
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