Theorem lssmex 14368

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 4858	. . . 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 14366	. . . . 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 4809	. . . 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 2298	. . . 4 |- ( U e. S <-> U e. ( LSubSp ` W ) )
7	6	biimpi 120	. . 3 |- ( U e. S -> U e. ( LSubSp ` W ) )
8		relelfvdm 5671	. . 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 2816	1 |- ( U e. S -> W e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   ~Pcpw 3652   |-> cmpt 4150   dom cdm 4725   Rel wrel 4730   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159  Scalarcsca 13162   .scvsca 13163   LSubSpclss 14365

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-dm 4735  df-iota 5286  df-fv 5334  df-lssm 14366

This theorem is referenced by:  islssm  14370