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Theorem iswspthn 41047
 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 (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))
Distinct variable groups:   𝑓,𝐺   𝑓,𝑊
Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem iswspthn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 breq2 4587 . . 3 (𝑤 = 𝑊 → (𝑓(SPathS‘𝐺)𝑤𝑓(SPathS‘𝐺)𝑊))
21exbidv 1837 . 2 (𝑤 = 𝑊 → (∃𝑓 𝑓(SPathS‘𝐺)𝑤 ↔ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))
3 wspthsn 41046 . 2 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}
42, 3elrab2 3333 1 (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  SPathScspths 40920   WWalkSN cwwlksn 41029   WSPathsN cwwspthsn 41031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wwlksn 41034  df-wspthsn 41036 This theorem is referenced by:  wspthnp  41048  wspthsnwspthsnon  41122  fusgreg2wsp  41500
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