Theorem lssmex 14629

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 ∈ 𝑆 → 𝑊 ∈ 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 4888	. . . 4 ⊢ Rel (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)})
2		df-lssm 14627	. . . . 5 ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)})
3	2	releqi 4838	. . . 4 ⊢ (Rel LSubSp ↔ Rel (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)}))
4	1, 3	mpbir 146	. . 3 ⊢ Rel LSubSp
5		lssmex.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
6	5	eleq2i 2301	. . . 4 ⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ (LSubSp‘𝑊))
7	6	biimpi 120	. . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ (LSubSp‘𝑊))
8		relelfvdm 5707	. . 3 ⊢ ((Rel LSubSp ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ dom LSubSp)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ dom LSubSp)
10	9	elexd 2829	1 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815  𝒫 cpw 3674   ↦ cmpt 4176  dom cdm 4754  Rel wrel 4759  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  Scalarcsca 13377   ·_𝑠 cvsca 13378  LSubSpclss 14626

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-dm 4764  df-iota 5317  df-fv 5365  df-lssm 14627

This theorem is referenced by:  islssm  14631