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Mirrors > Home > MPE Home > Th. List > relwlk | Structured version Visualization version GIF version |
Description: The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.) |
Ref | Expression |
---|---|
relwlk | ⊢ Rel (Walks‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wlks 27380 | . 2 ⊢ Walks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}) | |
2 | 1 | relmptopab 7394 | 1 ⊢ Rel (Walks‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: if-wif 1057 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 {csn 4566 {cpr 4568 dom cdm 5554 Rel wrel 5559 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 + caddc 10539 ...cfz 12891 ..^cfzo 13032 ♯chash 13689 Word cword 13860 Vtxcvtx 26780 iEdgciedg 26781 Walkscwlks 27377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-wlks 27380 |
This theorem is referenced by: wlkop 27408 istrl 27477 isclwlk 27553 usgrgt2cycl 32377 |
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