Theorem lssmex 14776

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 4908	. . . 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 14774	. . . . 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 4858	. . . 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 2305	. . . 4 ⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ (LSubSp‘𝑊))
7	6	biimpi 120	. . 3 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ (LSubSp‘𝑊))
8		relelfvdm 5727	. . 3 ⊢ ((Rel LSubSp ∧ 𝑈 ∈ (LSubSp‘𝑊)) → 𝑊 ∈ dom LSubSp)
9	4, 7, 8	sylancr 418	. 2 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ dom LSubSp)
10	9	elexd 2835	1 ⊢ (𝑈 ∈ 𝑆 → 𝑊 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821  𝒫 cpw 3688   ↦ cmpt 4192  dom cdm 4774  Rel wrel 4779  ‘cfv 5377  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  Scalarcsca 13487   ·_𝑠 cvsca 13488  LSubSpclss 14773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-dm 4784  df-iota 5337  df-fv 5385  df-lssm 14774

This theorem is used by:  islssm  14778