Theorem reldmfrlm 43830

Description: The domain of the free module function is a relation. (Contributed by SN, 24-Sep-2026.)

Assertion

Ref	Expression
reldmfrlm	⊢ Rel dom freeLMod

Proof of Theorem reldmfrlm

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frlm 22108	. 2 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
2	1	reldmmpo 7584	1 ⊢ Rel dom freeLMod

This proof depends on syntax axioms:  Vcvv 3454  {csn 4588   × cxp 5656  dom cdm 5658  Rel wrel 5663  ‘cfv 6566  (class class class)co 7451  ringLModcrglmod 21498   ⊕_m cdsmm 22092   freeLMod cfrlm 22107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-oprab 7455  df-mpo 7456  df-frlm 22108

This theorem is used by:  frlmvscl  43832