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Theorem islinds 21231
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 3462 . . . . 5 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
2 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
32pweqd 4578 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 (Baseβ€˜π‘Š))
4 breq2 5110 . . . . . . 7 (𝑀 = π‘Š β†’ (( I β†Ύ 𝑠) LIndF 𝑀 ↔ ( I β†Ύ 𝑠) LIndF π‘Š))
53, 4rabeqbidv 3423 . . . . . 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 21229 . . . . . 6 LIndS = (𝑀 ∈ V ↦ {𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ∣ ( I β†Ύ 𝑠) LIndF 𝑀})
7 fvex 6856 . . . . . . . 8 (Baseβ€˜π‘Š) ∈ V
87pwex 5336 . . . . . . 7 𝒫 (Baseβ€˜π‘Š) ∈ V
98rabex 5290 . . . . . 6 {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š} ∈ V
105, 6, 9fvmpt 6949 . . . . 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 2820 . . 3 (π‘Š ∈ 𝑉 β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ∣ ( I β†Ύ 𝑠) LIndF π‘Š}))
13 reseq2 5933 . . . . 5 (𝑠 = 𝑋 β†’ ( I β†Ύ 𝑠) = ( I β†Ύ 𝑋))
1413breq1d 5116 . . . 4 (𝑠 = 𝑋 β†’ (( I β†Ύ 𝑠) LIndF π‘Š ↔ ( I β†Ύ 𝑋) LIndF π‘Š))
1514elrab 3646 . . 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 5303 . . . 4 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
18 islinds.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
1918sseq2i 3974 . . . 4 (𝑋 βŠ† 𝐡 ↔ 𝑋 βŠ† (Baseβ€˜π‘Š))
2017, 19bitr4i 278 . . 3 (𝑋 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑋 βŠ† 𝐡)
2120anbi1i 625 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   βŠ† wss 3911  π’« cpw 4561   class class class wbr 5106   I cid 5531   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17088   LIndF clindf 21226  LIndSclinds 21227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-linds 21229
This theorem is referenced by:  linds1  21232  linds2  21233  islinds2  21235  lindsss  21246  lindsmm  21250  lsslinds  21253  islinds5  32203  lindspropd  32218
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