Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isconngr Structured version   Visualization version   GIF version

Theorem isconngr 41351
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 41349 . . 3 ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
21eleq2i 2679 . 2 (𝐺 ∈ ConnGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝})
3 fvex 6098 . . . . . 6 (Vtx‘𝑔) ∈ V
4 id 22 . . . . . . 7 (𝑣 = (Vtx‘𝑔) → 𝑣 = (Vtx‘𝑔))
5 raleq 3114 . . . . . . 7 (𝑣 = (Vtx‘𝑔) → (∀𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
64, 5raleqbidv 3128 . . . . . 6 (𝑣 = (Vtx‘𝑔) → (∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝))
73, 6sbcie 3436 . . . . 5 ([(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝)
87abbii 2725 . . . 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}
98eleq2i 2679 . . 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})
10 fveq2 6088 . . . . . 6 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isconngr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2661 . . . . 5 ( = 𝐺 → (Vtx‘) = 𝑉)
13 fveq2 6088 . . . . . . . . 9 ( = 𝐺 → (PathsOn‘) = (PathsOn‘𝐺))
1413oveqd 6544 . . . . . . . 8 ( = 𝐺 → (𝑘(PathsOn‘)𝑛) = (𝑘(PathsOn‘𝐺)𝑛))
1514breqd 4588 . . . . . . 7 ( = 𝐺 → (𝑓(𝑘(PathsOn‘)𝑛)𝑝𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
16152exbidv 1838 . . . . . 6 ( = 𝐺 → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1712, 16raleqbidv 3128 . . . . 5 ( = 𝐺 → (∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
1812, 17raleqbidv 3128 . . . 4 ( = 𝐺 → (∀𝑘 ∈ (Vtx‘)∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
19 fveq2 6088 . . . . . 6 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
20 fveq2 6088 . . . . . . . . . 10 (𝑔 = → (PathsOn‘𝑔) = (PathsOn‘))
2120oveqd 6544 . . . . . . . . 9 (𝑔 = → (𝑘(PathsOn‘𝑔)𝑛) = (𝑘(PathsOn‘)𝑛))
2221breqd 4588 . . . . . . . 8 (𝑔 = → (𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝𝑓(𝑘(PathsOn‘)𝑛)𝑝))
23222exbidv 1838 . . . . . . 7 (𝑔 = → (∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2419, 23raleqbidv 3128 . . . . . 6 (𝑔 = → (∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝 ↔ ∀𝑛 ∈ (Vtx‘)∃𝑓𝑝 𝑓(𝑘(PathsOn‘)𝑛)𝑝))
2519, 24raleqbidv 3128 . . . . 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))
2625cbvabv 2733 . . . 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}
2718, 26elab2g 3321 . . 3 (𝐺𝑊 → (𝐺 ∈ {𝑔 ∣ ∀𝑘 ∈ (Vtx‘𝑔)∀𝑛 ∈ (Vtx‘𝑔)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
289, 27syl5bb 270 . 2 (𝐺𝑊 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
292, 28syl5bb 270 1 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wral 2895  [wsbc 3401   class class class wbr 4577  cfv 5790  (class class class)co 6527  Vtxcvtx 40224  PathsOncpthson 40916  ConnGraphcconngr 41348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-conngr 41349
This theorem is referenced by:  0conngr  41354  0vconngr  41355  1conngr  41356  conngrv2edg  41357
  Copyright terms: Public domain W3C validator