Theorem isspth 30134

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 30132	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		cnveq 5861	. . . 4 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃)
3	2	funeqd 6562	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃))
4	3	adantl 487	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃))
5		reltrls 30104	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 6990	1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   class class class wbr 5111  ^◡ccnv 5662  Fun wfun 6534  ‘cfv 6540  Trailsctrls 30100  SPathscspths 30123

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-trls 30102  df-spths 30127

This theorem is used by:  spthispth  30136  spthdifv  30146  spthdep  30147  pthdepisspth  30148  spthonepeq  30165  uhgrwkspth  30168  usgr2wlkspth  30172  usgr2pth  30177  spthcycl  30219  2spthd  30357  0spth  30544  3spthd  30598  upgrimspths  48733