MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ispthson Structured version   Visualization version   GIF version
Theorem ispthson 29508
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 29506 . . 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 5152 . 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 5146 . . . 4 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (𝑓(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑝 ↔ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃))
5 breq12 5146 . . . 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 2726 . . 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 5527 . 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 509 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 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  {copab 5203  β€˜cfv 6537  (class class class)co 7405  Vtxcvtx 28764  TrailsOnctrlson 29457  Pathscpths 29478  PathsOncpthson 29480
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-pthson 29484
This theorem is referenced by:  pthsonprop  29510  pthonpth  29514  spthonpthon  29517  0pthon  29889  1pthond  29906  3pthond  29937
  Copyright terms: Public domain W3C validator