Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  upgrspths1wlk Structured version   Visualization version   GIF version

Theorem upgrspths1wlk 40944
 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 40928 . 2 (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝)})
2 vex 3176 . . . . . . . 8 𝑓 ∈ V
3 vex 3176 . . . . . . . 8 𝑝 ∈ V
4 isTrl 40904 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝑓 ∈ V ∧ 𝑝 ∈ V) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
52, 3, 4mp3an23 1408 . . . . . . 7 (𝐺 ∈ UPGraph → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
65adantr 480 . . . . . 6 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(TrailS‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
7 upgrwlkdvde 40943 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ 𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝) → Fun 𝑓)
873exp 1256 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝑓(1Walks‘𝐺)𝑝 → (Fun 𝑝 → Fun 𝑓)))
98com23 84 . . . . . . . 8 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(1Walks‘𝐺)𝑝 → Fun 𝑓)))
109imp 444 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(1Walks‘𝐺)𝑝 → Fun 𝑓))
1110pm4.71d 664 . . . . . 6 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(1Walks‘𝐺)𝑝 ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑓)))
126, 11bitr4d 270 . . . . 5 ((𝐺 ∈ UPGraph ∧ Fun 𝑝) → (𝑓(TrailS‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑝))
1312ex 449 . . . 4 (𝐺 ∈ UPGraph → (Fun 𝑝 → (𝑓(TrailS‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑝)))
1413pm5.32rd 670 . . 3 (𝐺 ∈ UPGraph → ((𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝) ↔ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)))
1514opabbidv 4648 . 2 (𝐺 ∈ UPGraph → {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝐺)𝑝 ∧ Fun 𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
161, 15eqtrd 2644 1 (𝐺 ∈ UPGraph → (SPathS‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ Fun 𝑝)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642  ◡ccnv 5037  Fun wfun 5798  ‘cfv 5804   UPGraph cupgr 25747  1Walksc1wlks 40796  TrailSctrls 40899  SPathScspths 40920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-1wlks 40800  df-wlks 40801  df-trls 40901  df-spths 40924 This theorem is referenced by:  upgrwlkdvspth  40945
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