Theorem pthsonfval 30359

Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)

Hypothesis

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

Assertion

Ref	Expression
pthsonfval	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)})

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

Proof of Theorem pthsonfval

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 29613	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ V)
3	2	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
4		simpl 488	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	4, 1	eleqtrdi 2874	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 490	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2874	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		df-pthson 30334	. 2 ⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
9	3, 5, 7, 8	mptmpoopabovd 8123	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  {copab 5171  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  TrailsOnctrlson 30307  Pathscpths 30328  PathsOncpthson 30330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-pthson 30334

This theorem is used by:  ispthson  30361