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Theorem isconngr 27953
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 27951 . . 3 ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
21eleq2i 2903 . 2 (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
3 fvex 6656 . . . . . 6 (Vtx‘𝑔) ∈ V
4 raleq 3390 . . . . . . 7 (𝑣 = (Vtx‘𝑔) → (∀𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
54raleqbi1dv 3388 . . . . . 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 3789 . . . . 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 2886 . . . 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 2903 . . 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 6643 . . . . . 6 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
10 isconngr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
119, 10syl6eqr 2874 . . . . 5 ( = 𝐺 → (Vtx‘) = 𝑉)
12 fveq2 6643 . . . . . . . . 9 ( = 𝐺 → (PathsOn‘) = (PathsOn‘𝐺))
1312oveqd 7147 . . . . . . . 8 ( = 𝐺 → (𝑘(PathsOn‘)𝑛) = (𝑘(PathsOn‘𝐺)𝑛))
1413breqd 5050 . . . . . . 7 ( = 𝐺 → (𝑓(𝑘(PathsOn‘)𝑛)𝑝𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
15142exbidv 1926 . . . . . 6 ( = 𝐺 → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1611, 15raleqbidv 3386 . . . . 5 ( = 𝐺 → (∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1711, 16raleqbidv 3386 . . . 4 ( = 𝐺 → (∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
18 fveq2 6643 . . . . . 6 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
19 fveq2 6643 . . . . . . . . . 10 (𝑔 = → (PathsOn‘𝑔) = (PathsOn‘))
2019oveqd 7147 . . . . . . . . 9 (𝑔 = → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘)𝑛))
2120breqd 5050 . . . . . . . 8 (𝑔 = → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝𝑓(𝑘(PathsOn‘)𝑛)𝑝))
22212exbidv 1926 . . . . . . 7 (𝑔 = → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2318, 22raleqbidv 3386 . . . . . 6 (𝑔 = → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2418, 23raleqbidv 3386 . . . . 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 2889 . . . 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 3645 . . 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 286 . 2 (𝐺𝑊 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
282, 27syl5bb 286 1 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wex 1781  wcel 2115  {cab 2799  wral 3126  [wsbc 3749   class class class wbr 5039  cfv 6328  (class class class)co 7130  Vtxcvtx 26768  PathsOncpthson 27482  ConnGraphcconngr 27950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-conngr 27951
This theorem is referenced by:  0conngr  27956  0vconngr  27957  1conngr  27958  conngrv2edg  27959
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