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Theorem isspth 29742
Description: Conditions for a pair of classes/functions to be a simple path (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
isspth (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))

Proof of Theorem isspth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spthsfval 29740 . 2 (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝)}
2 cnveq 5884 . . . 4 (𝑝 = 𝑃𝑝 = 𝑃)
32funeqd 6588 . . 3 (𝑝 = 𝑃 → (Fun 𝑝 ↔ Fun 𝑃))
43adantl 481 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun 𝑝 ↔ Fun 𝑃))
5 reltrls 29712 . 2 Rel (Trails‘𝐺)
61, 4, 5brfvopabrbr 7013 1 (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540   class class class wbr 5143  ccnv 5684  Fun wfun 6555  cfv 6561  Trailsctrls 29708  SPathscspths 29731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-trls 29710  df-spths 29735
This theorem is referenced by:  spthispth  29744  spthdifv  29753  spthdep  29754  pthdepisspth  29755  spthonepeq  29772  uhgrwkspth  29775  usgr2wlkspth  29779  usgr2pth  29784  2spthd  29961  0spth  30145  3spthd  30195  spthcycl  35134
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