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Mirrors > Home > MPE Home > Th. List > upgrspthswlk | Structured version Visualization version GIF version |
Description: The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
upgrspthswlk | ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spthsfval 29754 | . 2 ⊢ (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} | |
2 | istrl 29728 | . . . . . 6 ⊢ (𝑓(Trails‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)) | |
3 | upgrwlkdvde 29769 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝) → Fun ◡𝑓) | |
4 | 3 | 3exp 1118 | . . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → (Fun ◡𝑝 → Fun ◡𝑓))) |
5 | 4 | com23 86 | . . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓))) |
6 | 5 | imp 406 | . . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 → Fun ◡𝑓)) |
7 | 6 | pm4.71d 561 | . . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Walks‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓))) |
8 | 2, 7 | bitr4id 290 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ Fun ◡𝑝) → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝)) |
9 | 8 | ex 412 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (Fun ◡𝑝 → (𝑓(Trails‘𝐺)𝑝 ↔ 𝑓(Walks‘𝐺)𝑝))) |
10 | 9 | pm5.32rd 578 | . . 3 ⊢ (𝐺 ∈ UPGraph → ((𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝))) |
11 | 10 | opabbidv 5213 | . 2 ⊢ (𝐺 ∈ UPGraph → {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
12 | 1, 11 | eqtrid 2786 | 1 ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 {copab 5209 ◡ccnv 5687 Fun wfun 6556 ‘cfv 6562 UPGraphcupgr 29111 Walkscwlks 29628 Trailsctrls 29722 SPathscspths 29745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-edg 29079 df-uhgr 29089 df-upgr 29113 df-wlks 29631 df-trls 29724 df-spths 29749 |
This theorem is referenced by: upgrwlkdvspth 29771 |
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