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Mirrors > Home > MPE Home > Th. List > rellindf | Structured version Visualization version GIF version |
Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5605. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
rellindf | ⊢ Rel LIndF |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lindf 20768 | . 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 5690 | 1 ⊢ Rel LIndF |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 [wsbc 3694 ∖ cdif 3863 {csn 4541 dom cdm 5551 “ cima 5554 Rel wrel 5556 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Scalarcsca 16805 ·𝑠 cvsca 16806 0gc0g 16944 LSpanclspn 20008 LIndF clindf 20766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-opab 5116 df-xp 5557 df-rel 5558 df-lindf 20768 |
This theorem is referenced by: lindff 20777 lindfind 20778 f1lindf 20784 lindfmm 20789 lsslindf 20792 |
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