Theorem iscycl 30360

Description: Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)

Assertion

Ref	Expression
iscycl	⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem iscycl

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

Step	Hyp	Ref	Expression
1		cycls 30358	. 2 ⊢ (Cycles‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
2		fveq1 6876	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
3	2	adantl 487	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘0) = (𝑃‘0))
4		simpr 490	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑝 = 𝑃)
5		fveq2 6877	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
6	5	adantr 486	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (♯‘𝑓) = (♯‘𝐹))
7	4, 6	fveq12d 6884	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑝‘(♯‘𝑓)) = (𝑃‘(♯‘𝐹)))
8	3, 7	eqeq12d 2777	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
9		relpths 30285	. 2 ⊢ Rel (Paths‘𝐺)
10	1, 8, 9	brfvopabrbr 6982	1 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   class class class wbr 5103  ‘cfv 6531  0cc0 11181  ♯chash 14454  Pathscpths 30277  Cyclesccycls 30354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-pths 30281  df-cycls 30356

This theorem is used by:  cyclprop  30362  subgrcycl  30367  cycliscrct  30369  cyclnumvtx  30370  cyclnspth  30371  pthisspthorcycl  30372  spthcycl  30374  cyclispthon  30375  0cycl  30707  lp1cycl  30725  2cycld  30727  3cycld  30761  upgrimcycls  48953  gpgprismgr4cycllem11  49147