Theorem reldmfrlm 43671

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 22060	. 2 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
2	1	reldmmpo 7556	1 ⊢ Rel dom freeLMod

This proof depends on syntax axioms:  Vcvv 3450  {csn 4584   × cxp 5648  dom cdm 5650  Rel wrel 5655  ‘cfv 6541  (class class class)co 7422  ringLModcrglmod 21454   ⊕_m cdsmm 22044   freeLMod cfrlm 22059

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-oprab 7426  df-mpo 7427  df-frlm 22060

This theorem is used by:  frlmvscl  43673