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Theorem frlmrcl 18597
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 18585 . . 3  |- freeLMod  =  ( r  e.  _V , 
i  e.  _V  |->  ( r  (+)m  ( i  X.  {
(ringLMod `  r ) } ) ) )
43reldmmpt2 6398 . 2  |-  Rel  dom freeLMod
51, 2, 4strov2rcl 14542 1  |-  ( X  e.  B  ->  R  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   ` cfv 5588  (class class class)co 6285   Basecbs 14493  ringLModcrglmod 17627    (+)m cdsmm 18569   freeLMod cfrlm 18584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-slot 14497  df-base 14498  df-frlm 18585
This theorem is referenced by:  frlmbasfsupp  18598  frlmbassup  18599  frlmbasmap  18600  frlmvscafval  18606
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