Theorem isconngr 27533

Description: The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)

Hypothesis

Ref	Expression
isconngr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
isconngr	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))

Distinct variable groups:   𝑓,𝑘,𝑛,𝑝,𝐺   𝑘,𝑉,𝑛

Allowed substitution hints:   𝑉(𝑓,𝑝)   𝑊(𝑓,𝑘,𝑛,𝑝)

Proof of Theorem isconngr

Dummy variables 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-conngr 27531	. . 3 ⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
2	1	eleq2i 2870	. 2 ⊢ (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
3		fvex 6424	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
4		raleq 3321	. . . . . . 7 ⊢ (𝑣 = (Vtx‘𝑔) → (∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
5	4	raleqbi1dv 3329	. . . . . 6 ⊢ (𝑣 = (Vtx‘𝑔) → (∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
6	3, 5	sbcie 3668	. . . . 5 ⊢ ([(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝)
7	6	abbii 2916	. . . 4 ⊢ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
8	7	eleq2i 2870	. . 3 ⊢ (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ 𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
9		fveq2 6411	. . . . . 6 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = (Vtx‘𝐺))
10		isconngr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
11	9, 10	syl6eqr 2851	. . . . 5 ⊢ (ℎ = 𝐺 → (Vtx‘ℎ) = 𝑉)
12		fveq2 6411	. . . . . . . . 9 ⊢ (ℎ = 𝐺 → (PathsOn‘ℎ) = (PathsOn‘𝐺))
13	12	oveqd 6895	. . . . . . . 8 ⊢ (ℎ = 𝐺 → (𝑘(PathsOn‘ℎ)𝑛) = (𝑘(PathsOn‘𝐺)𝑛))
14	13	breqd 4854	. . . . . . 7 ⊢ (ℎ = 𝐺 → (𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
15	14	2exbidv 2020	. . . . . 6 ⊢ (ℎ = 𝐺 → (∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
16	11, 15	raleqbidv 3335	. . . . 5 ⊢ (ℎ = 𝐺 → (∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
17	11, 16	raleqbidv 3335	. . . 4 ⊢ (ℎ = 𝐺 → (∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
18		fveq2 6411	. . . . . 6 ⊢ (𝑔 = ℎ → (Vtx‘𝑔) = (Vtx‘ℎ))
19		fveq2 6411	. . . . . . . . . 10 ⊢ (𝑔 = ℎ → (PathsOn‘𝑔) = (PathsOn‘ℎ))
20	19	oveqd 6895	. . . . . . . . 9 ⊢ (𝑔 = ℎ → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘ℎ)𝑛))
21	20	breqd 4854	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝))
22	21	2exbidv 2020	. . . . . . 7 ⊢ (𝑔 = ℎ → (∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝))
23	18, 22	raleqbidv 3335	. . . . . 6 ⊢ (𝑔 = ℎ → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝))
24	18, 23	raleqbidv 3335	. . . . 5 ⊢ (𝑔 = ℎ → (∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝))
25	24	cbvabv 2924	. . . 4 ⊢ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = {ℎ ∣ ∀𝑘 ∈ (Vtx‘ℎ)∀𝑛 ∈ (Vtx‘ℎ)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘ℎ)𝑛)𝑝}
26	17, 25	elab2g 3545	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
27	8, 26	syl5bb 275	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
28	2, 27	syl5bb 275	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653  ∃wex 1875   ∈ wcel 2157  {cab 2785  ∀wral 3089  [wsbc 3633   class class class wbr 4843  ‘cfv 6101  (class class class)co 6878  Vtxcvtx 26231  PathsOncpthson 26968  ConnGraphcconngr 27530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-conngr 27531

This theorem is referenced by:  0conngr  27536  0vconngr  27537  1conngr  27538  conngrv2edg  27539