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Theorem islinds 20196
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 (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))

Proof of Theorem islinds
Dummy variables 𝑠 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3243 . . . . 5 (𝑊𝑉𝑊 ∈ V)
2 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
32pweqd 4196 . . . . . . 7 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊))
4 breq2 4689 . . . . . . 7 (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊))
53, 4rabeqbidv 3226 . . . . . 6 (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
6 df-linds 20194 . . . . . 6 LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
7 fvex 6239 . . . . . . . 8 (Base‘𝑊) ∈ V
87pwex 4878 . . . . . . 7 𝒫 (Base‘𝑊) ∈ V
98rabex 4845 . . . . . 6 {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V
105, 6, 9fvmpt 6321 . . . . 5 (𝑊 ∈ V → (LIndS‘𝑊) = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
111, 10syl 17 . . . 4 (𝑊𝑉 → (LIndS‘𝑊) = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})
1211eleq2d 2716 . . 3 (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}))
13 reseq2 5423 . . . . 5 (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋))
1413breq1d 4695 . . . 4 (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊))
1514elrab 3396 . . 3 (𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
1612, 15syl6bb 276 . 2 (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
177elpw2 4858 . . . 4 (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊))
18 islinds.b . . . . 5 𝐵 = (Base‘𝑊)
1918sseq2i 3663 . . . 4 (𝑋𝐵𝑋 ⊆ (Base‘𝑊))
2017, 19bitr4i 267 . . 3 (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋𝐵)
2120anbi1i 731 . 2 ((𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))
2216, 21syl6bb 276 1 (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191   class class class wbr 4685   I cid 5052  cres 5145  cfv 5926  Basecbs 15904   LIndF clindf 20191  LIndSclinds 20192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-linds 20194
This theorem is referenced by:  linds1  20197  linds2  20198  islinds2  20200  lindsss  20211  lindsmm  20215  lsslinds  20218
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