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Theorem pths 25864
Description: The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Assertion
Ref Expression
pths ((𝑉𝑋𝐸𝑌) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Distinct variable groups:   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝
Allowed substitution hints:   𝑋(𝑓,𝑝)   𝑌(𝑓,𝑝)

Proof of Theorem pths
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . 2 (𝑉𝑋𝑉 ∈ V)
2 elex 3089 . 2 (𝐸𝑌𝐸 ∈ V)
3 df-pth 25806 . . . 4 Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
43a1i 11 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}))
5 oveq12 6434 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 Trails 𝑒) = (𝑉 Trails 𝐸))
65adantl 480 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑣 Trails 𝑒) = (𝑉 Trails 𝐸))
76breqd 4492 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑓(𝑣 Trails 𝑒)𝑝𝑓(𝑉 Trails 𝐸)𝑝))
873anbi1d 1394 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
98opabbidv 4546 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
10 simpl 471 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
11 simpr 475 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
12 3anass 1034 . . . . . 6 ((𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
1312a1i 11 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))))
1413opabbidv 4546 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))})
15 trliswlk 25837 . . . . 5 (𝑓(𝑉 Trails 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
1615wlkres 25818 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))} ∈ V)
1714, 16eqeltrd 2592 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} ∈ V)
184, 9, 10, 11, 17ovmpt2d 6561 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
191, 2, 18syl2an 492 1 ((𝑉𝑋𝐸𝑌) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  Vcvv 3077  cin 3443  c0 3777  {cpr 4030   class class class wbr 4481  {copab 4540  ccnv 4931  cres 4934  cima 4935  Fun wfun 5683  cfv 5689  (class class class)co 6425  cmpt2 6427  0cc0 9689  1c1 9690  ..^cfzo 12199  #chash 12844   Trails ctrail 25795   Paths cpath 25796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-oadd 7325  df-er 7503  df-map 7620  df-pm 7621  df-en 7716  df-dom 7717  df-sdom 7718  df-fin 7719  df-card 8522  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-n0 11046  df-z 11117  df-uz 11424  df-fz 12063  df-fzo 12200  df-hash 12845  df-word 13009  df-wlk 25804  df-trail 25805  df-pth 25806
This theorem is referenced by:  ispth  25866
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