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Theorem pthsonfval 28396
Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
pthsonfval.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
pthsonfval ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)})
Distinct variable groups:   𝑓,𝐺,𝑝   𝐴,𝑓,𝑝   𝐡,𝑓,𝑝   𝑓,𝑉,𝑝

Proof of Theorem pthsonfval
Dummy variables π‘Ž 𝑏 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsonfval.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
211vgrex 27661 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐺 ∈ V)
32adantr 481 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐺 ∈ V)
4 simpl 483 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
54, 1eleqtrdi 2847 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ (Vtxβ€˜πΊ))
6 simpr 485 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ 𝑉)
76, 1eleqtrdi 2847 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ (Vtxβ€˜πΊ))
8 df-pthson 28374 . 2 PathsOn = (𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(π‘Ž(TrailsOnβ€˜π‘”)𝑏)𝑝 ∧ 𝑓(Pathsβ€˜π‘”)𝑝)}))
93, 5, 7, 8mptmpoopabovd 7990 1 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441   class class class wbr 5092  {copab 5154  β€˜cfv 6479  (class class class)co 7337  Vtxcvtx 27655  TrailsOnctrlson 28347  Pathscpths 28368  PathsOncpthson 28370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-pthson 28374
This theorem is referenced by:  ispthson  28398
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