Theorem lssmex 13851

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 4790	. . . 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 13849	. . . . 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 4742	. . . 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 2260	. . . 4 |- ( U e. S <-> U e. ( LSubSp ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. S -> U e. ( LSubSp ` W ) )
8		relelfvdm 5586	. . 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 2773	1 |- ( U e. S -> W e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760   ~Pcpw 3601   |-> cmpt 4090   dom cdm 4659   Rel wrel 4664   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695  Scalarcsca 12698   .scvsca 12699   LSubSpclss 13848

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-dm 4669  df-iota 5215  df-fv 5262  df-lssm 13849

This theorem is referenced by:  islssm  13853