Theorem isspth 29744

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.

Step	Hyp	Ref	Expression
1		spthsfval 29742	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		cnveq 5820	. . . 4 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃)
3	2	funeqd 6512	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃))
4	3	adantl 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃))
5		reltrls 29715	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 6936	1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   class class class wbr 5096  ^◡ccnv 5621  Fun wfun 6484  ‘cfv 6490  Trailsctrls 29711  SPathscspths 29733

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 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-trls 29713  df-spths 29737

This theorem is referenced by:  spthispth  29746  spthdifv  29755  spthdep  29756  pthdepisspth  29757  spthonepeq  29774  uhgrwkspth  29777  usgr2wlkspth  29781  usgr2pth  29786  2spthd  29963  0spth  30150  3spthd  30200  spthcycl  35272  upgrimspths  48098