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Theorem ispthson 29676
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 29674 . . 3 ((𝐴𝑉𝐵𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)})
32breqd 5156 . 2 ((𝐴𝑉𝐵𝑉) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹{⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)}𝑃))
4 breq12 5150 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃))
5 breq12 5150 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓(Paths‘𝐺)𝑝𝐹(Paths‘𝐺)𝑃))
64, 5anbi12d 630 . . 3 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝) ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))
7 eqid 2726 . . 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 5532 . 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 508 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 205  wa 394   = wceq 1534  wcel 2099   class class class wbr 5145  {copab 5207  cfv 6546  (class class class)co 7416  Vtxcvtx 28929  TrailsOnctrlson 29625  Pathscpths 29646  PathsOncpthson 29648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-pthson 29652
This theorem is referenced by:  pthsonprop  29678  pthonpth  29682  spthonpthon  29685  0pthon  30057  1pthond  30074  3pthond  30105
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