Theorem spthsfval 29702

Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)

Assertion

Ref	Expression
spthsfval	⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem spthsfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 262	. 2 ⊢ (𝑔 = 𝐺 → (Fun ◡𝑝 ↔ Fun ◡𝑝))
2		df-spths 29697	. 2 ⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})
3	1, 2	fvmptopab 7409	1 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}

Syntax hints:   ∧ wa 395   = wceq 1541   class class class wbr 5095  {copab 5157  ^◡ccnv 5620  Fun wfun 6482  ‘cfv 6488  Trailsctrls 29671  SPathscspths 29693

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-spths 29697

This theorem is referenced by:  isspth  29704  upgrspthswlk  29720