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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 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 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 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-conngr 27531 . . 3 ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
21eleq2i 2870 . 2 (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
3 fvex 6424 . . . . . 6 (Vtx‘𝑔) ∈ V
4 raleq 3321 . . . . . . 7 (𝑣 = (Vtx‘𝑔) → (∀𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
54raleqbi1dv 3329 . . . . . 6 (𝑣 = (Vtx‘𝑔) → (∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
63, 5sbcie 3668 . . . . 5 ([(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝)
76abbii 2916 . . . 4 {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
87eleq2i 2870 . . 3 (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ 𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
9 fveq2 6411 . . . . . 6 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
10 isconngr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
119, 10syl6eqr 2851 . . . . 5 ( = 𝐺 → (Vtx‘) = 𝑉)
12 fveq2 6411 . . . . . . . . 9 ( = 𝐺 → (PathsOn‘) = (PathsOn‘𝐺))
1312oveqd 6895 . . . . . . . 8 ( = 𝐺 → (𝑘(PathsOn‘)𝑛) = (𝑘(PathsOn‘𝐺)𝑛))
1413breqd 4854 . . . . . . 7 ( = 𝐺 → (𝑓(𝑘(PathsOn‘)𝑛)𝑝𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
15142exbidv 2020 . . . . . 6 ( = 𝐺 → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1611, 15raleqbidv 3335 . . . . 5 ( = 𝐺 → (∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1711, 16raleqbidv 3335 . . . 4 ( = 𝐺 → (∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
18 fveq2 6411 . . . . . 6 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fveq2 6411 . . . . . . . . . 10 (𝑔 = → (PathsOn‘𝑔) = (PathsOn‘))
2019oveqd 6895 . . . . . . . . 9 (𝑔 = → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘)𝑛))
2120breqd 4854 . . . . . . . 8 (𝑔 = → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝𝑓(𝑘(PathsOn‘)𝑛)𝑝))
22212exbidv 2020 . . . . . . 7 (𝑔 = → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2318, 22raleqbidv 3335 . . . . . 6 (𝑔 = → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2418, 23raleqbidv 3335 . . . . 5 (𝑔 = → (∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2524cbvabv 2924 . . . 4 {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} = { ∣ ∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝}
2617, 25elab2g 3545 . . 3 (𝐺𝑊 → (𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
278, 26syl5bb 275 . 2 (𝐺𝑊 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
282, 27syl5bb 275 1 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
Colors of variables: wff setvar class
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
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