Theorem linds2 21745

Description: An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Assertion

Ref	Expression
linds2	⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)

Proof of Theorem linds2

Step	Hyp	Ref	Expression
1		elfvdm 6934	. . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
2		eqid 2728	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
3	2	islinds 21743	. . . 4 ⊢ (𝑊 ∈ dom LIndS → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
4	1, 3	syl 17	. . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
5	4	ibi 267	. 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
6	5	simprd 495	1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099   ⊆ wss 3947   class class class wbr 5148   I cid 5575  dom cdm 5678   ↾ cres 5680  ‘cfv 6548  Basecbs 17180   LIndF clindf 21738  LIndSclinds 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-linds 21741

This theorem is referenced by:  lindsind2  21753  lindsss  21758  f1linds  21759