Theorem spthson 28109

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 = (Vtx‘𝐺)

Assertion

Ref	Expression
spthson	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(SPathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(SPaths‘𝐺)𝑝)})

Distinct variable groups:   𝑓,𝐺,𝑝   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝑓,𝑉,𝑝

Proof of Theorem spthson

Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pthsonfval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	1vgrex 27372	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 483	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	eleqtrdi 2849	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 485	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2849	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		df-spthson 28087	. 2 ⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
9	3, 5, 7, 8	mptmpoopabovd 7922	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(SPathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(SPaths‘𝐺)𝑝)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  {copab 5136  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366  TrailsOnctrlson 28059  SPathscspths 28081  SPathsOncspthson 28083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-spthson 28087

This theorem is referenced by:  isspthson  28111