Theorem linds2 20637

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 6570	. . . 4 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
2		eqid 2795	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
3	2	islinds 20635	. . . 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 268	. 2 ⊢ (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
6	5	simprd 496	1 ⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2081   ⊆ wss 3859   class class class wbr 4962   I cid 5347  dom cdm 5443   ↾ cres 5445  ‘cfv 6225  Basecbs 16312   LIndF clindf 20630  LIndSclinds 20631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-res 5455  df-iota 6189  df-fun 6227  df-fv 6233  df-linds 20633

This theorem is referenced by:  lindsind2  20645  lindsss  20650  f1linds  20651