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Theorem pthsonprop 29510
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 29508 . . 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 1163 . 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 29484 . 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 29470 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Vtxcvtx 28764  TrailsOnctrlson 29457  Pathscpths 29478  PathsOncpthson 29480
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-pthson 29484
This theorem is referenced by:  pthonispth  29512  pthontrlon  29513
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