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Theorem frlmrcl 27203
 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 freeLMod
frlmrcl.b
Assertion
Ref Expression
frlmrcl

Proof of Theorem frlmrcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmval.f . 2 freeLMod
2 frlmrcl.b . 2
3 df-frlm 27192 . . 3 freeLMod m ringLMod
43reldmmpt2 6182 . 2 freeLMod
51, 2, 4strov2rcl 16632 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  cvv 2957  csn 3815   cxp 4877  cfv 5455  (class class class)co 6082  cbs 13470  ringLModcrglmod 16242   m cdsmm 27175   freeLMod cfrlm 27190 This theorem is referenced by:  frlmbassup  27204  frlmbasmap  27205  frlmvscafval  27208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-slot 13474  df-base 13475  df-frlm 27192
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