Theorem islinds 20515

Description: Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypothesis

Ref	Expression
islinds.b	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
islinds	⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))

Proof of Theorem islinds

Dummy variables 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3429	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6433	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3	2	pweqd 4383	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
4		breq2 4877	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊))
5	3, 4	rabeqbidv 3408	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
6		df-linds 20513	. . . . . 6 ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
7		fvex 6446	. . . . . . . 8 ⊢ (Base‘𝑊) ∈ V
8	7	pwex 5080	. . . . . . 7 ⊢ 𝒫 (Base‘𝑊) ∈ V
9	8	rabex 5037	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V
10	5, 6, 9	fvmpt 6529	. . . . 5 ⊢ (𝑊 ∈ V → (LIndS‘𝑊) = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
11	1, 10	syl 17	. . . 4 ⊢ (𝑊 ∈ 𝑉 → (LIndS‘𝑊) = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
12	11	eleq2d 2892	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}))
13		reseq2 5624	. . . . 5 ⊢ (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋))
14	13	breq1d 4883	. . . 4 ⊢ (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊))
15	14	elrab 3585	. . 3 ⊢ (𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
16	12, 15	syl6bb 279	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
17	7	elpw2 5050	. . . 4 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊))
18		islinds.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
19	18	sseq2i 3855	. . . 4 ⊢ (𝑋 ⊆ 𝐵 ↔ 𝑋 ⊆ (Base‘𝑊))
20	17, 19	bitr4i 270	. . 3 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ 𝐵)
21	20	anbi1i 619	. 2 ⊢ ((𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))
22	16, 21	syl6bb 279	1 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3121  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4378   class class class wbr 4873   I cid 5249   ↾ cres 5344  ‘cfv 6123  Basecbs 16222   LIndF clindf 20510  LIndSclinds 20511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-res 5354  df-iota 6086  df-fun 6125  df-fv 6131  df-linds 20513

This theorem is referenced by:  linds1  20516  linds2  20517  islinds2  20519  lindsss  20530  lindsmm  20534  lsslinds  20537