Theorem istrl 28942

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 28941	. 2 ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)}
2		cnveq 5871	. . . 4 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
3	2	funeqd 6567	. . 3 ⊢ (𝑓 = 𝐹 → (Fun ◡𝑓 ↔ Fun ◡𝐹))
4	3	adantr 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ◡𝑓 ↔ Fun ◡𝐹))
5		relwlk 28872	. 2 ⊢ Rel (Walks‘𝐺)
6	1, 4, 5	brfvopabrbr 6992	1 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   class class class wbr 5147  ^◡ccnv 5674  Fun wfun 6534  ‘cfv 6540  Walkscwlks 28842  Trailsctrls 28936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-wlks 28845  df-trls 28938

This theorem is referenced by:  trliswlk  28943  trlf1  28944  trlres  28946  upgristrl  28948  2pthnloop  28977  upgrspthswlk  28984  uhgrwkspth  29001  usgr2wlkspth  29005  uspgrn2crct  29051  crctcshtrl  29066  2trld  29181  0trl  29364  1trld  29384  ntrl2v2e  29400  3trld  29414  iseupthf1o  29444  subgrtrl  34112