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Theorem ispthson 29576
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 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
ispthson (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍)) β†’ (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)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 = (Vtxβ€˜πΊ)
21pthsonfval 29574 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐴(PathsOnβ€˜πΊ)𝐡) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)})
32breqd 5163 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) β†’ (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹{βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)}𝑃))
4 breq12 5157 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ↔ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃))
5 breq12 5157 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑓(Pathsβ€˜πΊ)𝑝 ↔ 𝐹(Pathsβ€˜πΊ)𝑃))
64, 5anbi12d 630 . . 3 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ ((𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝) ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
7 eqid 2728 . . 3 {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)} = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)}
86, 7brabga 5540 . 2 ((𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍) β†’ (𝐹{βŸ¨π‘“, π‘βŸ© ∣ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ∧ 𝑓(Pathsβ€˜πΊ)𝑝)}𝑃 ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
93, 8sylan9bb 508 1 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ π‘ˆ ∧ 𝑃 ∈ 𝑍)) β†’ (𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐹(Pathsβ€˜πΊ)𝑃)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  {copab 5214  β€˜cfv 6553  (class class class)co 7426  Vtxcvtx 28829  TrailsOnctrlson 29525  Pathscpths 29546  PathsOncpthson 29548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-pthson 29552
This theorem is referenced by:  pthsonprop  29578  pthonpth  29582  spthonpthon  29585  0pthon  29957  1pthond  29974  3pthond  30005
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