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Theorem islinds 21750
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 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
islinds (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))

Proof of Theorem islinds
Dummy variables 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3492 . . . . 5 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6902 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
32pweqd 4623 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 (Baseβ€˜π‘Š))
4 breq2 5156 . . . . . . 7 (𝑀 = π‘Š β†’ (( I β†Ύ 𝑠) LIndF 𝑀 ↔ ( I β†Ύ 𝑠) LIndF π‘Š))
53, 4rabeqbidv 3448 . . . . . 6 (𝑀 = π‘Š β†’ {𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ∣ ( I β†Ύ 𝑠) LIndF 𝑀} = {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š})
6 df-linds 21748 . . . . . 6 LIndS = (𝑀 ∈ V ↦ {𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ∣ ( I β†Ύ 𝑠) LIndF 𝑀})
7 fvex 6915 . . . . . . . 8 (Baseβ€˜π‘Š) ∈ V
87pwex 5384 . . . . . . 7 𝒫 (Baseβ€˜π‘Š) ∈ V
98rabex 5338 . . . . . 6 {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š} ∈ V
105, 6, 9fvmpt 7010 . . . . 5 (π‘Š ∈ V β†’ (LIndSβ€˜π‘Š) = {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š})
111, 10syl 17 . . . 4 (π‘Š ∈ 𝑉 β†’ (LIndSβ€˜π‘Š) = {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š})
1211eleq2d 2815 . . 3 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š}))
13 reseq2 5984 . . . . 5 (𝑠 = 𝑋 β†’ ( I β†Ύ 𝑠) = ( I β†Ύ 𝑋))
1413breq1d 5162 . . . 4 (𝑠 = 𝑋 β†’ (( I β†Ύ 𝑠) LIndF π‘Š ↔ ( I β†Ύ 𝑋) LIndF π‘Š))
1514elrab 3684 . . 3 (𝑋 ∈ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š} ↔ (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š))
1612, 15bitrdi 286 . 2 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
177elpw2 5351 . . . 4 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
18 islinds.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
1918sseq2i 4011 . . . 4 (𝑋 βŠ† 𝐡 ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
2017, 19bitr4i 277 . . 3 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† 𝐡)
2120anbi1i 622 . 2 ((𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š))
2216, 21bitrdi 286 1 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   βŠ† wss 3949  π’« cpw 4606   class class class wbr 5152   I cid 5579   β†Ύ cres 5684  β€˜cfv 6553  Basecbs 17187   LIndF clindf 21745  LIndSclinds 21746
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-linds 21748
This theorem is referenced by:  linds1  21751  linds2  21752  islinds2  21754  lindsss  21765  lindsmm  21769  lsslinds  21772  islinds5  33103  lindspropd  33123
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