Theorem ispthson 29877

Description: Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)

Hypothesis

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

Assertion

Ref	Expression
ispthson	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))

Proof of Theorem ispthson

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

Step	Hyp	Ref	Expression
1		pthsonfval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	pthsonfval 29875	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)})
3	2	breqd 5101	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ 𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)}𝑃))
4		breq12 5095	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ↔ 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃))
5		breq12 5095	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓(Paths‘𝐺)𝑝 ↔ 𝐹(Paths‘𝐺)𝑃))
6	4, 5	anbi12d 640	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝) ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))
7		eqid 2752	. . 3 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)}
8	6, 7	brabga 5494	. 2 ⊢ ((𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)}𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))
9	3, 8	sylan9bb 516	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132   class class class wbr 5090  {copab 5152  ‘cfv 6506  (class class class)co 7381  Vtxcvtx 29132  TrailsOnctrlson 29825  Pathscpths 29845  PathsOncpthson 29847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-pthson 29851

This theorem is referenced by:  pthsonprop  29879  pthonpth  29883  spthonpthon  29886  0pthon  30264  1pthond  30281  3pthond  30312