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Theorem pthsonprop 28998
Description: Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
Hypothesis
Ref Expression
pthsonfval.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
pthsonprop (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))

Proof of Theorem pthsonprop
Dummy variables π‘Ž 𝑏 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsonfval.v . 2 𝑉 = (Vtxβ€˜πΊ)
21ispthson 28996 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
323adantl1 1166 . 2 (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) β†’ (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
4 df-pthson 28972 . 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)}))
51, 3, 4wksonproplem 28958 1 (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Vtxcvtx 28253  TrailsOnctrlson 28945  Pathscpths 28966  PathsOncpthson 28968
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-pthson 28972
This theorem is referenced by:  pthonispth  29000  pthontrlon  29001
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