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Theorem istrl 29624

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 29623	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ^◡𝑓)}
2		cnveq 5837	. . . 4 ⊢ (𝑓 = 𝐹 → ^◡𝑓 = ^◡𝐹)
3	2	funeqd 6538	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
4	3	adantr 480	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
5		relwlk 29554	. 2 ⊢ Rel (Walks‘𝐺)
6	1, 4, 5	brfvopabrbr 6965	1 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 class class class wbr 5107 ^◡ccnv 5637 Fun wfun 6505 ‘cfv 6511 Walkscwlks 29524 Trailsctrls 29618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fv 6519 df-wlks 29527 df-trls 29620

This theorem is referenced by: trliswlk 29625 trlf1 29626 trlres 29628 upgristrl 29630 dfpth2 29659 2pthnloop 29661 upgrspthswlk 29668 uhgrwkspth 29685 usgr2wlkspth 29689 uspgrn2crct 29738 crctcshtrl 29753 2trld 29868 0trl 30051 1trld 30071 ntrl2v2e 30087 3trld 30101 iseupthf1o 30131 subgrtrl 35120 upgrimtrls 47906 gpgprismgr4cycllem11 48095

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