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Theorem isconngr 30122
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 30120 . . 3 ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
21eleq2i 2818 . 2 (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
3 fvex 6914 . . . . . 6 (Vtx‘𝑔) ∈ V
4 raleq 3312 . . . . . . 7 (𝑣 = (Vtx‘𝑔) → (∀𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
54raleqbi1dv 3323 . . . . . 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 3820 . . . . 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 2796 . . . 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 2818 . . 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 6901 . . . . . 6 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
10 isconngr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
119, 10eqtr4di 2784 . . . . 5 ( = 𝐺 → (Vtx‘) = 𝑉)
12 fveq2 6901 . . . . . . . . 9 ( = 𝐺 → (PathsOn‘) = (PathsOn‘𝐺))
1312oveqd 7441 . . . . . . . 8 ( = 𝐺 → (𝑘(PathsOn‘)𝑛) = (𝑘(PathsOn‘𝐺)𝑛))
1413breqd 5164 . . . . . . 7 ( = 𝐺 → (𝑓(𝑘(PathsOn‘)𝑛)𝑝𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
15142exbidv 1920 . . . . . 6 ( = 𝐺 → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1611, 15raleqbidv 3330 . . . . 5 ( = 𝐺 → (∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1711, 16raleqbidv 3330 . . . 4 ( = 𝐺 → (∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
18 fveq2 6901 . . . . . 6 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fveq2 6901 . . . . . . . . . 10 (𝑔 = → (PathsOn‘𝑔) = (PathsOn‘))
2019oveqd 7441 . . . . . . . . 9 (𝑔 = → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘)𝑛))
2120breqd 5164 . . . . . . . 8 (𝑔 = → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝𝑓(𝑘(PathsOn‘)𝑛)𝑝))
22212exbidv 1920 . . . . . . 7 (𝑔 = → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2318, 22raleqbidv 3330 . . . . . 6 (𝑔 = → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2418, 23raleqbidv 3330 . . . . 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 2799 . . . 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 3668 . . 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, 26bitrid 282 . 2 (𝐺𝑊 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
282, 27bitrid 282 1 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wral 3051  [wsbc 3776   class class class wbr 5153  cfv 6554  (class class class)co 7424  Vtxcvtx 28932  PathsOncpthson 29651  ConnGraphcconngr 30119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-ov 7427  df-conngr 30120
This theorem is referenced by:  0conngr  30125  0vconngr  30126  1conngr  30127  conngrv2edg  30128
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