Theorem trlsonfval 29787

Description: The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)

Hypothesis

Ref	Expression
trlsonfval.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
trlsonfval	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)})

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

Proof of Theorem trlsonfval

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

Step	Hyp	Ref	Expression
1		trlsonfval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	1vgrex 29085	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	eleqtrdi 2847	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2847	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		df-trlson 29775	. 2 ⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
9	3, 5, 7, 8	mptmpoopabovd 8027	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  {copab 5148  ‘cfv 6492  (class class class)co 7360  Vtxcvtx 29079  WalksOncwlkson 29681  Trailsctrls 29772  TrailsOnctrlson 29773

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-trlson 29775

This theorem is referenced by:  istrlson  29788