Theorem isspth 29486

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 29484	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
2		cnveq 5866	. . . 4 ⊢ (𝑝 = 𝑃 → ◡𝑝 = ◡𝑃)
3	2	funeqd 6563	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ◡𝑝 ↔ Fun ◡𝑃))
4	3	adantl 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑝 ↔ Fun ◡𝑃))
5		reltrls 29456	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 6988	1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   class class class wbr 5141  ^◡ccnv 5668  Fun wfun 6530  ‘cfv 6536  Trailsctrls 29452  SPathscspths 29475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-trls 29454  df-spths 29479

This theorem is referenced by:  spthispth  29488  spthdifv  29495  spthdep  29496  pthdepisspth  29497  spthonepeq  29514  uhgrwkspth  29517  usgr2wlkspth  29521  usgr2pth  29526  2spthd  29700  0spth  29884  3spthd  29934  spthcycl  34648