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Theorem spthson 29503
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 28766 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐺 ∈ V)
32adantr 480 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐺 ∈ V)
4 simpl 482 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ 𝑉)
54, 1eleqtrdi 2837 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ (Vtxβ€˜πΊ))
6 simpr 484 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ 𝑉)
76, 1eleqtrdi 2837 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ 𝐡 ∈ (Vtxβ€˜πΊ))
8 df-spthson 29481 . 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 8065 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  {copab 5203  β€˜cfv 6536  (class class class)co 7404  Vtxcvtx 28760  TrailsOnctrlson 29453  SPathscspths 29475  SPathsOncspthson 29477
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-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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-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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-spthson 29481
This theorem is referenced by:  isspthson  29505
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