Theorem islinds 21789

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 3450	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6840	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3	2	pweqd 4558	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
4		breq2 5089	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊))
5	3, 4	rabeqbidv 3407	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
6		df-linds 21787	. . . . . 6 ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
7		fvex 6853	. . . . . . . 8 ⊢ (Base‘𝑊) ∈ V
8	7	pwex 5322	. . . . . . 7 ⊢ 𝒫 (Base‘𝑊) ∈ V
9	8	rabex 5280	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V
10	5, 6, 9	fvmpt 6947	. . . . 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 2822	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}))
13		reseq2 5939	. . . . 5 ⊢ (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋))
14	13	breq1d 5095	. . . 4 ⊢ (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊))
15	14	elrab 3634	. . 3 ⊢ (𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
16	12, 15	bitrdi 287	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
17	7	elpw2 5275	. . . 4 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊))
18		islinds.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
19	18	sseq2i 3951	. . . 4 ⊢ (𝑋 ⊆ 𝐵 ↔ 𝑋 ⊆ (Base‘𝑊))
20	17, 19	bitr4i 278	. . 3 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ 𝐵)
21	20	anbi1i 625	. 2 ⊢ ((𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))
22	16, 21	bitrdi 287	1 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ⊆ wss 3889  𝒫 cpw 4541   class class class wbr 5085   I cid 5525   ↾ cres 5633  ‘cfv 6498  Basecbs 17179   LIndF clindf 21784  LIndSclinds 21785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-linds 21787

This theorem is referenced by:  linds1  21790  linds2  21791  islinds2  21793  lindsss  21804  lindsmm  21808  lsslinds  21811  islinds5  33427  lindspropd  33443