Theorem isclwlk 29856

Description: A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

Assertion

Ref	Expression
isclwlk	⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem isclwlk

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlks 29855	. 2 ⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2		fveq1 6833	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
3	2	adantl 481	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4		simpr 484	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
5		fveq2 6834	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
6	5	adantr 480	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
7	4, 6	fveq12d 6841	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
8	3, 7	eqeq12d 2753	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9		relwlk 29709	. 2 ⊢ Rel (Walks‘𝐺)
10	1, 8, 9	brfvopabrbr 6938	1 ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   class class class wbr 5086  ‘cfv 6492  0cc0 11029  ♯chash 14283  Walkscwlks 29680  ClWalkscclwlks 29853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-wlks 29683  df-clwlks 29854

This theorem is referenced by:  clwlkiswlk  29857  isclwlke  29860  isclwlkupgr  29861  clwlkcompbp  29865  clwlkl1loop  29866  crctisclwlk  29877  clwlkclwwlkflem  30089  clwlkclwwlkf  30093  0clwlk  30215