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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 3501 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
| 2 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
| 3 | 2 | pweqd 4617 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝑊)) |
| 4 | breq2 5147 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (( I ↾ 𝑠) LIndF 𝑤 ↔ ( I ↾ 𝑠) LIndF 𝑊)) | |
| 5 | 3, 4 | rabeqbidv 3455 | . . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤} = {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊}) |
| 6 | df-linds 21827 | . . . . . 6 ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) | |
| 7 | fvex 6919 | . . . . . . . 8 ⊢ (Base‘𝑊) ∈ V | |
| 8 | 7 | pwex 5380 | . . . . . . 7 ⊢ 𝒫 (Base‘𝑊) ∈ V |
| 9 | 8 | rabex 5339 | . . . . . 6 ⊢ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ∈ V |
| 10 | 5, 6, 9 | fvmpt 7016 | . . . . 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 2827 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ 𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊})) |
| 13 | reseq2 5992 | . . . . 5 ⊢ (𝑠 = 𝑋 → ( I ↾ 𝑠) = ( I ↾ 𝑋)) | |
| 14 | 13 | breq1d 5153 | . . . 4 ⊢ (𝑠 = 𝑋 → (( I ↾ 𝑠) LIndF 𝑊 ↔ ( I ↾ 𝑋) LIndF 𝑊)) |
| 15 | 14 | elrab 3692 | . . 3 ⊢ (𝑋 ∈ {𝑠 ∈ 𝒫 (Base‘𝑊) ∣ ( I ↾ 𝑠) LIndF 𝑊} ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
| 16 | 12, 15 | bitrdi 287 | . 2 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
| 17 | 7 | elpw2 5334 | . . . 4 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ (Base‘𝑊)) |
| 18 | islinds.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
| 19 | 18 | sseq2i 4013 | . . . 4 ⊢ (𝑋 ⊆ 𝐵 ↔ 𝑋 ⊆ (Base‘𝑊)) |
| 20 | 17, 19 | bitr4i 278 | . . 3 ⊢ (𝑋 ∈ 𝒫 (Base‘𝑊) ↔ 𝑋 ⊆ 𝐵) |
| 21 | 20 | anbi1i 624 | . 2 ⊢ ((𝑋 ∈ 𝒫 (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)) |
| 22 | 16, 21 | bitrdi 287 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 I cid 5577 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 LIndF clindf 21824 LIndSclinds 21825 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-linds 21827 |
| This theorem is referenced by: linds1 21830 linds2 21831 islinds2 21833 lindsss 21844 lindsmm 21848 lsslinds 21851 islinds5 33395 lindspropd 33411 |
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