Theorem iswspthn 29103

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 5153	. . 3 ⊢ (𝑤 = 𝑊 → (𝑓(SPaths‘𝐺)𝑤 ↔ 𝑓(SPaths‘𝐺)𝑊))
2	1	exbidv 1925	. 2 ⊢ (𝑤 = 𝑊 → (∃𝑓 𝑓(SPaths‘𝐺)𝑤 ↔ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
3		wspthsn 29102	. 2 ⊢ (𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
4	2, 3	elrab2 3687	1 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  SPathscspths 28970   WWalksN cwwlksn 29080   WSPathsN cwwspthsn 29082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-wwlksn 29085  df-wspthsn 29087

This theorem is referenced by:  wspthnp  29104  wspthsnwspthsnon  29170