Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2wlksotot Structured version   Visualization version   GIF version

Theorem el2wlksotot 26409
 Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
Assertion
Ref Expression
el2wlksotot (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝   𝑓,𝑋,𝑝   𝑓,𝑌,𝑝

Proof of Theorem el2wlksotot
Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2wlksot 26394 . . . 4 ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})
21adantr 480 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})
32eleq2d 2673 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑉 2WalksOt 𝐸) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)}))
4 eleq1 2676 . . . . . 6 (𝑡 = ⟨𝐴, 𝐵, 𝐶⟩ → (𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
542rexbidv 3039 . . . . 5 (𝑡 = ⟨𝐴, 𝐵, 𝐶⟩ → (∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
65elrab 3331 . . . 4 (⟨𝐴, 𝐵, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ↔ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
76a1i 11 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ↔ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))))
8 eqid 2610 . . . . . 6 𝐴, 𝐵, 𝐶⟩ = ⟨𝐴, 𝐵, 𝐶
9 otel3xp 5077 . . . . . 6 ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
108, 9mpan 702 . . . . 5 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
1110adantl 481 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
1211biantrurd 528 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))))
13 el2wlkonotot0 26399 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝑎𝑉𝑏𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ (𝐴 = 𝑎𝐶 = 𝑏 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
1413adantlr 747 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑎𝑉𝑏𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ (𝐴 = 𝑎𝐶 = 𝑏 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
15 simp3 1056 . . . . . 6 ((𝐴 = 𝑎𝐶 = 𝑏 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
1614, 15syl6bi 242 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝑎𝑉𝑏𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
1716rexlimdvva 3020 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
18 el2wlkonotot 26400 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
1918bicomd 212 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
20193adantr2 1214 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
21 simplr1 1096 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → 𝐴𝑉)
22 simplr3 1098 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → 𝐶𝑉)
23 simpr 476 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
24 oveq1 6556 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) = (𝐴(𝑉 2WalksOnOt 𝐸)𝑏))
2524eleq2d 2673 . . . . . . . 8 (𝑎 = 𝐴 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝑏)))
26 oveq2 6557 . . . . . . . . 9 (𝑏 = 𝐶 → (𝐴(𝑉 2WalksOnOt 𝐸)𝑏) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))
2726eleq2d 2673 . . . . . . . 8 (𝑏 = 𝐶 → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
2825, 27rspc2ev 3295 . . . . . . 7 ((𝐴𝑉𝐶𝑉 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))
2921, 22, 23, 28syl3anc 1318 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) → ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))
3029ex 449 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
3120, 30sylbid 229 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
3217, 31impbid 201 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (∃𝑎𝑉𝑏𝑉𝐴, 𝐵, 𝐶⟩ ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
337, 12, 323bitr2d 295 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
343, 33bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOt c2wlkot 26381   2WalksOnOt c2wlkonot 26382 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-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-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-ot 4134  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-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-wlk 26036  df-wlkon 26042  df-2wlkonot 26385  df-2wlksot 26386 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator