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Theorem isconngra 26200
 Description: The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
Assertion
Ref Expression
isconngra ((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
Distinct variable groups:   𝑓,𝑉,𝑘,𝑛,𝑝   𝑓,𝐸,𝑘,𝑛,𝑝
Allowed substitution hints:   𝑋(𝑓,𝑘,𝑛,𝑝)   𝑌(𝑓,𝑘,𝑛,𝑝)

Proof of Theorem isconngra
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
2 oveq12 6558 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 PathOn 𝑒) = (𝑉 PathOn 𝐸))
32oveqd 6566 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑘(𝑣 PathOn 𝑒)𝑛) = (𝑘(𝑉 PathOn 𝐸)𝑛))
43breqd 4594 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
542exbidv 1839 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
61, 5raleqbidv 3129 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∀𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
71, 6raleqbidv 3129 . 2 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
8 df-conngra 26198 . 2 ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}
97, 8brabga 4914 1 ((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  (class class class)co 6549   PathOn cpthon 26032   ConnGrph cconngra 26197 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-conngra 26198 This theorem is referenced by:  0conngra  26202  1conngra  26203
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