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Theorem iswspthn 26717
 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 4648 . . 3 (𝑤 = 𝑊 → (𝑓(SPaths‘𝐺)𝑤𝑓(SPaths‘𝐺)𝑊))
21exbidv 1848 . 2 (𝑤 = 𝑊 → (∃𝑓 𝑓(SPaths‘𝐺)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
3 wspthsn 26716 . 2 (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
42, 3elrab2 3360 1 (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1481  ∃wex 1702   ∈ wcel 1988   class class class wbr 4644  ‘cfv 5876  (class class class)co 6635  SPathscspths 26590   WWalksN cwwlksn 26699   WSPathsN cwwspthsn 26701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-wwlksn 26704  df-wspthsn 26706 This theorem is referenced by:  wspthnp  26718  wspthsnwspthsnon  26792
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