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Mirrors > Home > MPE Home > Th. List > islinds | Structured version Visualization version GIF version |
Description: Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
islinds.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
islinds | ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | fveq2 6645 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
3 | 2 | pweqd 4516 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊)) |
4 | breq2 5034 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊)) | |
5 | 3, 4 | rabeqbidv 3433 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}) |
6 | df-linds 20496 | . . . . . 6 ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) | |
7 | fvex 6658 | . . . . . . . 8 ⊢ (Base‘𝑊) ∈ V | |
8 | 7 | pwex 5246 | . . . . . . 7 ⊢ 𝒫 (Base‘𝑊) ∈ V |
9 | 8 | rabex 5199 | . . . . . 6 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V |
10 | 5, 6, 9 | fvmpt 6745 | . . . . 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 2875 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})) |
13 | reseq2 5813 | . . . . 5 ⊢ (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋)) | |
14 | 13 | breq1d 5040 | . . . 4 ⊢ (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊)) |
15 | 14 | elrab 3628 | . . 3 ⊢ (𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
16 | 12, 15 | syl6bb 290 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
17 | 7 | elpw2 5212 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊)) |
18 | islinds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
19 | 18 | sseq2i 3944 | . . . 4 ⊢ (𝑋 ⊆ 𝐵 ↔ 𝑋 ⊆ (Base‘𝑊)) |
20 | 17, 19 | bitr4i 281 | . . 3 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ 𝐵) |
21 | 20 | anbi1i 626 | . 2 ⊢ ((𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
22 | 16, 21 | syl6bb 290 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 I cid 5424 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 LIndF clindf 20493 LIndSclinds 20494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-linds 20496 |
This theorem is referenced by: linds1 20499 linds2 20500 islinds2 20502 lindsss 20513 lindsmm 20517 lsslinds 20520 islinds5 30983 lindspropd 30997 |
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