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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 5730. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
rellindf | ⊢ Rel LIndF |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lindf 21345 | . 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 5818 | 1 ⊢ Rel LIndF |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 [wsbc 3776 ∖ cdif 3944 {csn 4627 dom cdm 5675 “ cima 5678 Rel wrel 5680 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 Scalarcsca 17196 ·𝑠 cvsca 17197 0gc0g 17381 LSpanclspn 20570 LIndF clindf 21343 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3954 df-ss 3964 df-opab 5210 df-xp 5681 df-rel 5682 df-lindf 21345 |
This theorem is referenced by: lindff 21354 lindfind 21355 f1lindf 21361 lindfmm 21366 lsslindf 21369 |
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