Theorem frlmrcl 27202

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 27191	. . 3 |- freeLMod = ( r e. _V , i e. _V |-> ( r (+)m ( i X. { (ringLMod ` r ) } ) ) )
4	3	reldmmpt2 6181	. 2 |- Rel dom freeLMod
5	1, 2, 4	strov2rcl 16631	1 |- ( X e. B -> R e. _V )

Syntax hints:   -> wi 4   = wceq 1652   e. wcel 1725   _Vcvv 2956   {csn 3814   X. cxp 4876   ` cfv 5454  (class class class)co 6081   Basecbs 13469  ringLModcrglmod 16241   (+)_m cdsmm 27174   freeLMod cfrlm 27189

This theorem is referenced by:  frlmbassup  27203  frlmbasmap  27204  frlmvscafval  27207

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-slot 13473  df-base 13474  df-frlm 27191