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

Theorem iswspthn 29680
Description: An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
Assertion
Ref Expression
iswspthn (π‘Š ∈ (𝑁 WSPathsN 𝐺) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
Distinct variable groups:   𝑓,𝐺   𝑓,π‘Š
Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem iswspthn
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 breq2 5156 . . 3 (𝑀 = π‘Š β†’ (𝑓(SPathsβ€˜πΊ)𝑀 ↔ 𝑓(SPathsβ€˜πΊ)π‘Š))
21exbidv 1916 . 2 (𝑀 = π‘Š β†’ (βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀 ↔ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
3 wspthsn 29679 . 2 (𝑁 WSPathsN 𝐺) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀}
42, 3elrab2 3687 1 (π‘Š ∈ (𝑁 WSPathsN 𝐺) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  SPathscspths 29547   WWalksN cwwlksn 29657   WSPathsN cwwspthsn 29659
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-sbc 3779  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-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-ov 7429  df-oprab 7430  df-mpo 7431  df-wwlksn 29662  df-wspthsn 29664
This theorem is referenced by:  wspthnp  29681  wspthsnwspthsnon  29747
  Copyright terms: Public domain W3C validator