istrl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > istrl		Structured version Visualization version GIF version

Theorem istrl 30272

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 30271	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ^◡𝑓)}
2		cnveq 5851	. . . 4 ⊢ (𝑓 = 𝐹 → ^◡𝑓 = ^◡𝐹)
3	2	funeqd 6559	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
4	3	adantr 486	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑓 ↔ Fun ^◡𝐹))
5		relwlk 30199	. 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

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 class class class wbr 5103 ^◡ccnv 5650 Fun wfun 6531 ‘cfv 6537 Walkscwlks 30170 Trailsctrls 30266

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 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 6493 df-fun 6539 df-fv 6545 df-wlks 30173 df-trls 30268

This theorem is used by: trliswlk 30273 trlf1 30274 trlres 30276 upgristrl 30278 subgrtrl 30287 dfpth2 30307 2pthnloop 30310 upgrspthswlk 30317 uhgrwkspth 30334 usgr2wlkspth 30338 uspgrn2crct 30390 crctcshtrl 30405 2trld 30520 0trl 30706 1trld 30726 ntrl2v2e 30752 3trld 30766 iseupthf1o 30796 upgrimtrls 48973 gpgprismgr4cycllem11 49172

W3C validator