Theorem spthson 28217

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 27480	. . 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 2848	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺))
6		simpr 485	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉)
7	6, 1	eleqtrdi 2848	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ (Vtx‘𝐺))
8		df-spthson 28195	. 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 7965	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 1540   ∈ wcel 2105  Vcvv 3441   class class class wbr 5085  {copab 5147  ‘cfv 6463  (class class class)co 7313  Vtxcvtx 27474  TrailsOnctrlson 28167  SPathscspths 28189  SPathsOncspthson 28191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-spthson 28195

This theorem is referenced by:  isspthson  28219