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Theorem islinds 21699
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 3487 . . . . 5 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6884 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
32pweqd 4614 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 (Baseβ€˜π‘Š))
4 breq2 5145 . . . . . . 7 (𝑀 = π‘Š β†’ (( I β†Ύ 𝑠) LIndF 𝑀 ↔ ( I β†Ύ 𝑠) LIndF π‘Š))
53, 4rabeqbidv 3443 . . . . . 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 21697 . . . . . 6 LIndS = (𝑀 ∈ V ↦ {𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ∣ ( I β†Ύ 𝑠) LIndF 𝑀})
7 fvex 6897 . . . . . . . 8 (Baseβ€˜π‘Š) ∈ V
87pwex 5371 . . . . . . 7 𝒫 (Baseβ€˜π‘Š) ∈ V
98rabex 5325 . . . . . 6 {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š} ∈ V
105, 6, 9fvmpt 6991 . . . . 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 2813 . . 3 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š}))
13 reseq2 5969 . . . . 5 (𝑠 = 𝑋 β†’ ( I β†Ύ 𝑠) = ( I β†Ύ 𝑋))
1413breq1d 5151 . . . 4 (𝑠 = 𝑋 β†’ (( I β†Ύ 𝑠) LIndF π‘Š ↔ ( I β†Ύ 𝑋) LIndF π‘Š))
1514elrab 3678 . . 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 287 . 2 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
177elpw2 5338 . . . 4 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
18 islinds.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
1918sseq2i 4006 . . . 4 (𝑋 βŠ† 𝐡 ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
2017, 19bitr4i 278 . . 3 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† 𝐡)
2120anbi1i 623 . 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 287 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 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   βŠ† wss 3943  π’« cpw 4597   class class class wbr 5141   I cid 5566   β†Ύ cres 5671  β€˜cfv 6536  Basecbs 17150   LIndF clindf 21694  LIndSclinds 21695
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-linds 21697
This theorem is referenced by:  linds1  21700  linds2  21701  islinds2  21703  lindsss  21714  lindsmm  21718  lsslinds  21721  islinds5  32985  lindspropd  33004
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