Theorem reldmlmhm 20908

Description: Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Assertion

Ref	Expression
reldmlmhm	⊢ Rel dom LMHom

Proof of Theorem reldmlmhm

Dummy variables 𝑓 𝑠 𝑡 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lmhm 20905	. 2 ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))})
2	1	reldmmpo 7503	1 ⊢ Rel dom LMHom

Syntax hints:   ∧ wa 395   = wceq 1540  ∀wral 3044  {crab 3402  [wsbc 3750  dom cdm 5631  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Scalarcsca 17199   ·_𝑠 cvsca 17200   GrpHom cghm 19120  LModclmod 20742   LMHom clmhm 20902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-dm 5641  df-oprab 7373  df-mpo 7374  df-lmhm 20905

This theorem is referenced by:  mendbas  43142