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Theorem ispthson 29763
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 (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))

Proof of Theorem ispthson
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pthsonfval.v . . . 4 𝑉 = (Vtx‘𝐺)
21pthsonfval 29761 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)})
32breqd 5153 . 2 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)}𝑃))
4 breq12 5147 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃))
5 breq12 5147 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(Paths‘𝐺)𝑝𝐹(Paths‘𝐺)𝑃))
64, 5anbi12d 632 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝) ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))
7 eqid 2736 . . 3 {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)}
86, 7brabga 5538 . 2 ((𝐹𝑈𝑃𝑍) → (𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)}𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))
93, 8sylan9bb 509 1 (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107   class class class wbr 5142  {copab 5204  cfv 6560  (class class class)co 7432  Vtxcvtx 29014  TrailsOnctrlson 29710  Pathscpths 29731  PathsOncpthson 29733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-pthson 29737
This theorem is referenced by:  pthsonprop  29765  pthonpth  29769  spthonpthon  29772  0pthon  30147  1pthond  30164  3pthond  30195
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