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 ∈ 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 4856	. . . 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 14357	. . . . 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 4807	. . . 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 2296	. . . 4 ⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ (LSubSp‘𝑊))
7	6	biimpi 120	. . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ (LSubSp‘𝑊))
8		relelfvdm 5667	. . 3 ⊢ ((Rel LSubSp ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ dom LSubSp)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ dom LSubSp)
10	9	elexd 2814	1 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2800  𝒫 cpw 3650   ↦ cmpt 4148  dom cdm 4723  Rel wrel 4728  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  Scalarcsca 13153   ·_𝑠 cvsca 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