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Theorem upgrspthswlk 27527
 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.)
Assertion
Ref Expression
upgrspthswlk (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem upgrspthswlk
StepHypRef Expression
1 spthsfval 27511 . 2 (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝)}
2 upgrwlkdvde 27526 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ 𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝) → Fun 𝑓)
323exp 1116 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → (Fun 𝑝 → Fun 𝑓)))
43com23 86 . . . . . . . 8 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(Walks‘𝐺)𝑝 → Fun 𝑓)))
54imp 410 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(Walks‘𝐺)𝑝 → Fun 𝑓))
65pm4.71d 565 . . . . . 6 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(Walks‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
7 istrl 27486 . . . . . 6 (𝑓(Trails‘𝐺)𝑝 ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑓))
86, 7syl6rbbr 293 . . . . 5 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(Trails‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝))
98ex 416 . . . 4 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(Trails‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝)))
109pm5.32rd 581 . . 3 (𝐺 ∈ UPGraph → ((𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)))
1110opabbidv 5096 . 2 (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
121, 11syl5eq 2845 1 (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  {copab 5092  ◡ccnv 5518  Fun wfun 6318  ‘cfv 6324  UPGraphcupgr 26873  Walkscwlks 27386  Trailsctrls 27480  SPathscspths 27502 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-edg 26841  df-uhgr 26851  df-upgr 26875  df-wlks 27389  df-trls 27482  df-spths 27506 This theorem is referenced by:  upgrwlkdvspth  27528
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