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Theorem frlmrcl 20603
 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 20593 . . 3 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
43reldmmpo 7101 . 2 Rel dom freeLMod
51, 2, 4strov2rcl 16402 1 (𝑋𝐵𝑅 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2050  Vcvv 3416  {csn 4441   × cxp 5405  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  ringLModcrglmod 19663   ⊕m cdsmm 20577   freeLMod cfrlm 20592 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-slot 16343  df-base 16345  df-frlm 20593 This theorem is referenced by:  frlmbasfsupp  20604  frlmbasmap  20605  frlmvscafval  20612
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