|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 29617 | . 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 7683 | 1 ⊢ Rel (Walks‘𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: if-wif 1063 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 {csn 4626 {cpr 4628 dom cdm 5685 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 Vtxcvtx 29013 iEdgciedg 29014 Walkscwlks 29614 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-wlks 29617 | 
| This theorem is referenced by: wlkop 29646 istrl 29714 isclwlk 29793 usgrgt2cycl 35135 | 
| Copyright terms: Public domain | W3C validator |