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Theorem frlmrcl 20829
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 20819 . . 3 freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
43reldmmpo 7274 . 2 Rel dom freeLMod
51, 2, 4strov2rcl 16534 1 (𝑋𝐵𝑅 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557   × cxp 5546  cfv 6348  (class class class)co 7145  Basecbs 16471  ringLModcrglmod 19870  m cdsmm 20803   freeLMod cfrlm 20818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-slot 16475  df-base 16477  df-frlm 20819
This theorem is referenced by:  frlmbasfsupp  20830  frlmbasmap  20831  frlmvscafval  20838
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