Theorem isconngra 26200

Description: The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)

Assertion

Ref	Expression
isconngra	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 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.

Step	Hyp	Ref	Expression
1		simpl 472	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
2		oveq12 6558	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 PathOn 𝑒) = (𝑉 PathOn 𝐸))
3	2	oveqd 6566	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑘(𝑣 PathOn 𝑒)𝑛) = (𝑘(𝑉 PathOn 𝐸)𝑛))
4	3	breqd 4594	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
5	4	2exbidv 1839	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃𝑓∃𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
6	1, 5	raleqbidv 3129	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
7	1, 6	raleqbidv 3129	. 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))
8		df-conngra 26198	. 2 ⊢ ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}
9	7, 8	brabga 4914	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))

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