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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 brrelexi 5082. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
rellindf | ⊢ Rel LIndF |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lindf 19964 | . 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 5167 | 1 ⊢ Rel LIndF |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 [wsbc 3402 ∖ cdif 3537 {csn 4125 dom cdm 5038 “ cima 5041 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LSpanclspn 18792 LIndF clindf 19962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-lindf 19964 |
This theorem is referenced by: lindff 19973 lindfind 19974 f1lindf 19980 lindfmm 19985 lsslindf 19988 |
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