Theorem lssmex 14359

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 4856	. . . 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 14357	. . . . 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 4807	. . . 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 2296	. . . 4 |- ( U e. S <-> U e. ( LSubSp ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. S -> U e. ( LSubSp ` W ) )
8		relelfvdm 5667	. . 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 2814	1 |- ( U e. S -> W e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   {crab 2512   _Vcvv 2800   ~Pcpw 3650   |-> cmpt 4148   dom cdm 4723   Rel wrel 4728   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150  Scalarcsca 13153   .scvsca 13154   LSubSpclss 14356

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-xp 4729  df-rel 4730  df-dm 4733  df-iota 5284  df-fv 5332  df-lssm 14357

This theorem is referenced by:  islssm  14361