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Theorem isspth 29551
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(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))

Proof of Theorem isspth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spthsfval 29549 . 2 (SPathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ Fun ◑𝑝)}
2 cnveq 5876 . . . 4 (𝑝 = 𝑃 β†’ ◑𝑝 = ◑𝑃)
32funeqd 6575 . . 3 (𝑝 = 𝑃 β†’ (Fun ◑𝑝 ↔ Fun ◑𝑃))
43adantl 481 . 2 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (Fun ◑𝑝 ↔ Fun ◑𝑃))
5 reltrls 29521 . 2 Rel (Trailsβ€˜πΊ)
61, 4, 5brfvopabrbr 7002 1 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   class class class wbr 5148  β—‘ccnv 5677  Fun wfun 6542  β€˜cfv 6548  Trailsctrls 29517  SPathscspths 29540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-trls 29519  df-spths 29544
This theorem is referenced by:  spthispth  29553  spthdifv  29560  spthdep  29561  pthdepisspth  29562  spthonepeq  29579  uhgrwkspth  29582  usgr2wlkspth  29586  usgr2pth  29591  2spthd  29765  0spth  29949  3spthd  29999  spthcycl  34739
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