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Theorem upgrspths1wlk 40946
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.)
Assertion
Ref Expression
upgrspths1wlk (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem upgrspths1wlk
StepHypRef Expression
1 spthsfval 40930 . 2 (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝)})
2 vex 3175 . . . . . . . 8 𝑓 ∈ V
3 vex 3175 . . . . . . . 8 𝑝 ∈ V
4 isTrl 40906 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑓 ∈ V ∧ 𝑝 ∈ V) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
52, 3, 4mp3an23 1407 . . . . . . 7 (𝐺 ∈ UPGraph → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
65adantr 479 . . . . . 6 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
7 upgrwlkdvde 40945 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ 𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝) → Fun 𝑓)
873exp 1255 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑓(1Walks‘𝐺)𝑝 → (Fun 𝑝 → Fun 𝑓)))
98com23 83 . . . . . . . 8 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(1Walks‘𝐺)𝑝 → Fun 𝑓)))
109imp 443 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(1Walks‘𝐺)𝑝 → Fun 𝑓))
1110pm4.71d 663 . . . . . 6 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(1Walks‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
126, 11bitr4d 269 . . . . 5 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(TrailS‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑝))
1312ex 448 . . . 4 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(TrailS‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑝)))
1413pm5.32rd 669 . . 3 (𝐺 ∈ UPGraph → ((𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)))
1514opabbidv 4642 . 2 (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
161, 15eqtrd 2643 1 (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  Vcvv 3172   class class class wbr 4577  {copab 4636  ccnv 5027  Fun wfun 5784  cfv 5790   UPGraph cupgr 40308  1Walksc1wlks 40798  TrailSctrls 40901  SPathScspths 40922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-uhgr 40282  df-upgr 40310  df-edga 40354  df-1wlks 40802  df-wlks 40803  df-trls 40903  df-spths 40926
This theorem is referenced by:  upgrwlkdvspth  40947
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