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.

Step	Hyp	Ref	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 ) } ) ) )
4	3	reldmmpt2 6140	. 2 |- Rel dom freeLMod
5	1, 2, 4	strov2rcl 16586	1 |- ( X e. B -> R e. _V )

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