Theorem iswspthn 30135

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.

Step	Hyp	Ref	Expression
1		breq2 5114	. . 3 ⊢ (𝑤 = 𝑊 → (𝑓(SPaths‘𝐺)𝑤 ↔ 𝑓(SPaths‘𝐺)𝑊))
2	1	exbidv 1948	. 2 ⊢ (𝑤 = 𝑊 → (∃𝑓 𝑓(SPaths‘𝐺)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
3		wspthsn 30134	. 2 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
4	2, 3	elrab2 3663	1 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  SPathscspths 29997   WWalksN cwwlksn 30112   WSPathsN cwwspthsn 30114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-wwlksn 30117  df-wspthsn 30119

This theorem is referenced by:  wspthnp  30136  wspthsnwspthsnon  30202