MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isspth Structured version   Visualization version   GIF version

Theorem isspth 28380
Description: Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
Assertion
Ref Expression
isspth (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))

Proof of Theorem isspth
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 spthsfval 28378 . 2 (SPathsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ Fun ◑𝑝)}
2 cnveq 5815 . . . 4 (𝑝 = 𝑃 β†’ ◑𝑝 = ◑𝑃)
32funeqd 6506 . . 3 (𝑝 = 𝑃 β†’ (Fun ◑𝑝 ↔ Fun ◑𝑃))
43adantl 482 . 2 ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) β†’ (Fun ◑𝑝 ↔ Fun ◑𝑃))
5 reltrls 28350 . 2 Rel (Trailsβ€˜πΊ)
61, 4, 5brfvopabrbr 6928 1 (𝐹(SPathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun ◑𝑃))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540   class class class wbr 5092  β—‘ccnv 5619  Fun wfun 6473  β€˜cfv 6479  Trailsctrls 28346  SPathscspths 28369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fv 6487  df-trls 28348  df-spths 28373
This theorem is referenced by:  spthispth  28382  spthdifv  28389  spthdep  28390  pthdepisspth  28391  spthonepeq  28408  uhgrwkspth  28411  usgr2wlkspth  28415  usgr2pth  28420  2spthd  28594  0spth  28778  3spthd  28828  spthcycl  33390
  Copyright terms: Public domain W3C validator