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Theorem iswspthn 29607
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 5145 . . 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 29606 . 2 (𝑁 WSPathsN 𝐺) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀}
42, 3elrab2 3681 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 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  SPathscspths 29474   WWalksN cwwlksn 29584   WSPathsN cwwspthsn 29586
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-wwlksn 29589  df-wspthsn 29591
This theorem is referenced by:  wspthnp  29608  wspthsnwspthsnon  29674
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