Theorem islinds 21672

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 3485	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
2		fveq2 6881	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
3	2	pweqd 4611	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
4		breq2 5142	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊))
5	3, 4	rabeqbidv 3441	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
6		df-linds 21670	. . . . . 6 ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
7		fvex 6894	. . . . . . . 8 ⊢ (Base‘𝑊) ∈ V
8	7	pwex 5368	. . . . . . 7 ⊢ 𝒫 (Base‘𝑊) ∈ V
9	8	rabex 5322	. . . . . 6 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V
10	5, 6, 9	fvmpt 6988	. . . . 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 2811	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}))
13		reseq2 5966	. . . . 5 ⊢ (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋))
14	13	breq1d 5148	. . . 4 ⊢ (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊))
15	14	elrab 3675	. . 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 5335	. . . 4 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊))
18		islinds.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
19	18	sseq2i 4003	. . . 4 ⊢ (𝑋 ⊆ 𝐵 ↔ 𝑋 ⊆ (Base‘𝑊))
20	17, 19	bitr4i 278	. . 3 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ 𝐵)
21	20	anbi1i 623	. 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 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424  Vcvv 3466   ⊆ wss 3940  𝒫 cpw 4594   class class class wbr 5138   I cid 5563   ↾ cres 5668  ‘cfv 6533  Basecbs 17143   LIndF clindf 21667  LIndSclinds 21668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-iota 6485  df-fun 6535  df-fv 6541  df-linds 21670

This theorem is referenced by:  linds1  21673  linds2  21674  islinds2  21676  lindsss  21687  lindsmm  21691  lsslinds  21694  islinds5  32949  lindspropd  32968