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

Theorem trlsfval 29529
Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)
Assertion
Ref Expression
trlsfval (Trailsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ Fun ◑𝑓)}
Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem trlsfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 biidd 261 . 2 (𝑔 = 𝐺 β†’ (Fun ◑𝑓 ↔ Fun ◑𝑓))
2 df-trls 29526 . 2 Trails = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜π‘”)𝑝 ∧ Fun ◑𝑓)})
31, 2fvmptopab 7480 1 (Trailsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ Fun ◑𝑓)}
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1533   class class class wbr 5152  {copab 5214  β—‘ccnv 5681  Fun wfun 6547  β€˜cfv 6553  Walkscwlks 29430  Trailsctrls 29524
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-pr 5433
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-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  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-iota 6505  df-fun 6555  df-fv 6561  df-trls 29526
This theorem is referenced by:  istrl  29530  upgrtrls  29535
  Copyright terms: Public domain W3C validator