Theorem istrl 29608

Description: Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)

Assertion

Ref	Expression
istrl	⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))

Proof of Theorem istrl

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlsfval 29607	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)}
2		cnveq 5850	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
3	2	funeqd 6554	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ◡𝑓 ↔ Fun ◡𝐹))
4	3	adantr 480	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑓 ↔ Fun ◡𝐹))
5		relwlk 29538	. 2 ⊢ Rel (Walks‘𝐺)
6	1, 4, 5	brfvopabrbr 6979	1 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   class class class wbr 5116  ^◡ccnv 5650  Fun wfun 6521  ‘cfv 6527  Walkscwlks 29508  Trailsctrls 29602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fv 6535  df-wlks 29511  df-trls 29604

This theorem is referenced by:  trliswlk  29609  trlf1  29610  trlres  29612  upgristrl  29614  dfpth2  29643  2pthnloop  29645  upgrspthswlk  29652  uhgrwkspth  29669  usgr2wlkspth  29673  uspgrn2crct  29722  crctcshtrl  29737  2trld  29852  0trl  30035  1trld  30055  ntrl2v2e  30071  3trld  30085  iseupthf1o  30115  subgrtrl  35076