Theorem reldmfrlm 43544

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 22046	. 2 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
2	1	reldmmpo 7552	1 ⊢ Rel dom freeLMod

This proof depends on syntax axioms:  Vcvv 3451  {csn 4584   × cxp 5649  dom cdm 5651  Rel wrel 5656  ‘cfv 6537  (class class class)co 7418  ringLModcrglmod 21440   ⊕_m cdsmm 22030   freeLMod cfrlm 22045

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7422  df-mpo 7423  df-frlm 22046

This theorem is used by:  frlmvscl  43546