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Theorem frlmrcl 27093
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  |-  ( X  e.  B  ->  R  e.  _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 27082 . . 3  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
43reldmmpt2 6140 . 2  |-  Rel  dom freeLMod
51, 2, 4strov2rcl 16586 1  |-  ( X  e.  B  ->  R  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    X. cxp 4835   ` cfv 5413  (class class class)co 6040   Basecbs 13424  ringLModcrglmod 16196    (+)m cdsmm 27065   freeLMod cfrlm 27080
This theorem is referenced by:  frlmbassup  27094  frlmbasmap  27095  frlmvscafval  27098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-slot 13428  df-base 13429  df-frlm 27082
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