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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 5408. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
rellindf | ⊢ Rel LIndF |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lindf 20553 | . 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 | relopabi 5493 | 1 ⊢ Rel LIndF |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 386 ∈ wcel 2107 ∀wral 3090 [wsbc 3652 ∖ cdif 3789 {csn 4398 dom cdm 5357 “ cima 5360 Rel wrel 5362 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 Scalarcsca 16345 ·𝑠 cvsca 16346 0gc0g 16490 LSpanclspn 19370 LIndF clindf 20551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4951 df-xp 5363 df-rel 5364 df-lindf 20553 |
This theorem is referenced by: lindff 20562 lindfind 20563 f1lindf 20569 lindfmm 20574 lsslindf 20577 |
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