Theorem lssmex 13688

Description: If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)

Hypothesis

Ref	Expression
lssmex.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lssmex	|- ( U e. S -> W e. _V )

Proof of Theorem lssmex

Dummy variables a b j s w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptrel 4773	. . . 4 |- Rel ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
2		df-lssm 13686	. . . . 5 |- LSubSp = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
3	2	releqi 4727	. . . 4 |- ( Rel LSubSp <-> Rel ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } ) )
4	1, 3	mpbir 146	. . 3 |- Rel LSubSp
5		lssmex.s	. . . . 5 |- S = ( LSubSp ` W )
6	5	eleq2i 2256	. . . 4 |- ( U e. S <-> U e. ( LSubSp ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. S -> U e. ( LSubSp ` W ) )
8		relelfvdm 5566	. . 3 |- ( ( Rel LSubSp /\ U e. ( LSubSp ` W ) ) -> W e. dom LSubSp )
9	4, 7, 8	sylancr 414	. 2 |- ( U e. S -> W e. dom LSubSp )
10	9	elexd 2765	1 |- ( U e. S -> W e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   A.wral 2468   {crab 2472   _Vcvv 2752   ~Pcpw 3590   |-> cmpt 4079   dom cdm 4644   Rel wrel 4649   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592  Scalarcsca 12595   .scvsca 12596   LSubSpclss 13685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-xp 4650  df-rel 4651  df-dm 4654  df-iota 5196  df-fv 5243  df-lssm 13686

This theorem is referenced by:  islssm  13690