Theorem istrlson 30168

Description: Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)

Hypothesis

Ref	Expression
trlsonfval.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
istrlson	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))

Proof of Theorem istrlson

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlsonfval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	trlsonfval 30167	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)})
3	2	breqd 5114	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ 𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)}𝑃))
4		breq12 5108	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ↔ 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃))
5		breq12 5108	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(Trails‘𝐺)𝑝 ↔ 𝐹(Trails‘𝐺)𝑃))
6	4, 5	anbi12d 644	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝) ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
7		eqid 2760	. . 3 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)}
8	6, 7	brabga 5512	. 2 ⊢ ((𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)}𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
9	3, 8	sylan9bb 519	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  {copab 5167  ‘cfv 6533  (class class class)co 7413  Vtxcvtx 29453  WalksOncwlkson 30057  Trailsctrls 30152  TrailsOnctrlson 30153

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-trlson 30155

This theorem is used by:  trlsonprop  30169  trlontrl  30172  isspthonpth  30214  spthonepeq  30217  2trlond  30407  0trlon  30594  1pthond  30614  3trlond  30653