Theorem istrl 28109

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 28108	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)}
2		cnveq 5795	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
3	2	funeqd 6485	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ◡𝑓 ↔ Fun ◡𝐹))
4	3	adantr 482	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑓 ↔ Fun ◡𝐹))
5		relwlk 28038	. 2 ⊢ Rel (Walks‘𝐺)
6	1, 4, 5	brfvopabrbr 6904	1 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1539   class class class wbr 5081  ^◡ccnv 5599  Fun wfun 6452  ‘cfv 6458  Walkscwlks 28008  Trailsctrls 28103

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fv 6466  df-wlks 28011  df-trls 28105

This theorem is referenced by:  trliswlk  28110  trlf1  28111  trlres  28113  upgristrl  28115  2pthnloop  28144  upgrspthswlk  28151  uhgrwkspth  28168  usgr2wlkspth  28172  uspgrn2crct  28218  crctcshtrl  28233  2trld  28348  0trl  28531  1trld  28551  ntrl2v2e  28567  3trld  28581  iseupthf1o  28611  subgrtrl  33140