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Theorem isspth 28672
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 28670 . 2 (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝)}
2 cnveq 5829 . . . 4 (𝑝 = 𝑃𝑝 = 𝑃)
32funeqd 6523 . . 3 (𝑝 = 𝑃 → (Fun 𝑝 ↔ Fun 𝑃))
43adantl 482 . 2 ((𝑓 = 𝐹𝑝 = 𝑃) → (Fun 𝑝 ↔ Fun 𝑃))
5 reltrls 28642 . 2 Rel (Trails‘𝐺)
61, 4, 5brfvopabrbr 6945 1 (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541   class class class wbr 5105  ccnv 5632  Fun wfun 6490  cfv 6496  Trailsctrls 28638  SPathscspths 28661
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-trls 28640  df-spths 28665
This theorem is referenced by:  spthispth  28674  spthdifv  28681  spthdep  28682  pthdepisspth  28683  spthonepeq  28700  uhgrwkspth  28703  usgr2wlkspth  28707  usgr2pth  28712  2spthd  28886  0spth  29070  3spthd  29120  spthcycl  33723
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