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Theorem spthson 29573
Description: The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
pthsonfval.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
spthson ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(SPathsOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(SPathsβ€˜πΊ)𝑝)})
Distinct variable groups:   𝑓,𝐺,𝑝   𝐴,𝑓,𝑝   𝐡,𝑓,𝑝   𝑓,𝑉,𝑝

Proof of Theorem spthson
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 28833 . . 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 2838 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ (Vtxβ€˜πΊ))
6 simpr 483 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ 𝑉)
76, 1eleqtrdi 2838 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ (Vtxβ€˜πΊ))
8 df-spthson 29551 . 2 SPathsOn = (𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(π‘Ž(TrailsOnβ€˜π‘”)𝑏)𝑝 ∧ 𝑓(SPathsβ€˜π‘”)𝑝)}))
93, 5, 7, 8mptmpoopabovd 8091 1 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(SPathsOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(SPathsβ€˜πΊ)𝑝)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3471   class class class wbr 5150  {copab 5212  β€˜cfv 6551  (class class class)co 7424  Vtxcvtx 28827  TrailsOnctrlson 29523  SPathscspths 29545  SPathsOncspthson 29547
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 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-spthson 29551
This theorem is referenced by:  isspthson  29575
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