istrl - Metamath Proof Explorer

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Theorem istrl 29681

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 29680	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ^◡𝑓)}
2		cnveq 5858	. . . 4 ⊢ (𝑓 = 𝐹 → ^◡𝑓 = ^◡𝐹)
3	2	funeqd 6563	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
4	3	adantr 480	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
5		relwlk 29611	. 2 ⊢ Rel (Walks‘𝐺)
6	1, 4, 5	brfvopabrbr 6988	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 5124 ^◡ccnv 5658 Fun wfun 6530 ‘cfv 6536 Walkscwlks 29581 Trailsctrls 29675

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 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fv 6544 df-wlks 29584 df-trls 29677

This theorem is referenced by: trliswlk 29682 trlf1 29683 trlres 29685 upgristrl 29687 dfpth2 29716 2pthnloop 29718 upgrspthswlk 29725 uhgrwkspth 29742 usgr2wlkspth 29746 uspgrn2crct 29795 crctcshtrl 29810 2trld 29925 0trl 30108 1trld 30128 ntrl2v2e 30144 3trld 30158 iseupthf1o 30188 subgrtrl 35160 upgrimtrls 47886 gpgprismgr4cycllem11 48071

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