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| Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5741. (Contributed by Stefan O'Rear, 24-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| rellindf | ⊢ Rel LIndF | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-lindf 21826 | . 2 ⊢ LIndF = {〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel LIndF | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 ∖ cdif 3948 {csn 4626 dom cdm 5685 “ cima 5688 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LSpanclspn 20969 LIndF clindf 21824 | 
| 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-lindf 21826 | 
| This theorem is referenced by: lindff 21835 lindfind 21836 f1lindf 21842 lindfmm 21847 lsslindf 21850 | 
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