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Theorem frlmrcl 20149
Description: If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
Hypotheses
Ref Expression
frlmval.f 𝐹 = (𝑅 freeLMod 𝐼)
frlmrcl.b 𝐵 = (Base‘𝐹)
Assertion
Ref Expression
frlmrcl (𝑋𝐵𝑅 ∈ V)

Proof of Theorem frlmrcl
Dummy variables 𝑟 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2 𝐹 = (𝑅 freeLMod 𝐼)
2 frlmrcl.b . 2 𝐵 = (Base‘𝐹)
3 df-frlm 20139 . . 3 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
43reldmmpt2 6813 . 2 Rel dom freeLMod
51, 2, 4strov2rcl 15969 1 (𝑋𝐵𝑅 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210   × cxp 5141  cfv 5926  (class class class)co 6690  Basecbs 15904  ringLModcrglmod 19217  m cdsmm 20123   freeLMod cfrlm 20138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-slot 15908  df-base 15910  df-frlm 20139
This theorem is referenced by:  frlmbasfsupp  20150  frlmbasmap  20151  frlmvscafval  20157
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