Theorem istrlson 30092

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 30091	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)})
3	2	breqd 5122	. 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 5116	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ↔ 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃))
5		breq12 5116	. . . 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 2765	. . 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 5520	. 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 2146   class class class wbr 5111  {copab 5175  ‘cfv 6540  (class class class)co 7416  Vtxcvtx 29377  WalksOncwlkson 29981  Trailsctrls 30076  TrailsOnctrlson 30077

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-trlson 30079

This theorem is used by:  trlsonprop  30093  trlontrl  30096  isspthonpth  30138  spthonepeq  30141  2trlond  30331  0trlon  30518  1pthond  30538  3trlond  30571