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Theorem pthsonfval 29598
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 28859 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐺 ∈ V)
32adantr 479 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐺 ∈ V)
4 simpl 481 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
54, 1eleqtrdi 2835 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ (Vtxβ€˜πΊ))
6 simpr 483 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ 𝑉)
76, 1eleqtrdi 2835 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ (Vtxβ€˜πΊ))
8 df-pthson 29576 . 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 8085 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   class class class wbr 5143  {copab 5205  β€˜cfv 6543  (class class class)co 7416  Vtxcvtx 28853  TrailsOnctrlson 29549  Pathscpths 29570  PathsOncpthson 29572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-pthson 29576
This theorem is referenced by:  ispthson  29600
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