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Theorem cyclnumvtx 29785

Description: The number of vertices of a (non-trivial) cycle is the number of edges in the cycle. (Contributed by AV, 5-Oct-2025.)

**Assertion**
Ref	Expression
cyclnumvtx	⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran 𝑃) = (♯‘𝐹))

**Proof of Theorem cyclnumvtx**
Step	Hyp	Ref	Expression
1		iscycl 29776	. . . . 5 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		pthiswlk 29710	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2731	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 29602	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		wlkcl 29601	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
6		elnnnn0c 12432	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝐹)))
7		frel 6662	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Rel 𝑃)
8	7	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → Rel 𝑃)
9		fz1ssfz0 13529	. . . . . . . . . . . . . . . . 17 ⊢ (1...(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
10		fdm 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
11	9, 10	sseqtrrid 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (1...(♯‘𝐹)) ⊆ dom 𝑃)
12	11	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (1...(♯‘𝐹)) ⊆ dom 𝑃)
13	8, 12	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃))
14	10	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → dom 𝑃 = (0...(♯‘𝐹)))
15	14	difeq1d 4074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (dom 𝑃 ∖ (1...(♯‘𝐹))) = ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))))
16		nnnn0 12394	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
17		fz0sn0fz1 13551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (0...(♯‘𝐹)) = ({0} ∪ (1...(♯‘𝐹))))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) ∈ ℕ → (0...(♯‘𝐹)) = ({0} ∪ (1...(♯‘𝐹))))
19	18	difeq1d 4074	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ ℕ → ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))) = (({0} ∪ (1...(♯‘𝐹))) ∖ (1...(♯‘𝐹))))
20		1e0p1 12636	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 = (0 + 1)
21	20	oveq1i 7362	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...(♯‘𝐹)) = ((0 + 1)...(♯‘𝐹))
22	21	ineq2i 4166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({0} ∩ (1...(♯‘𝐹))) = ({0} ∩ ((0 + 1)...(♯‘𝐹)))
23	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ (1...(♯‘𝐹))) = ({0} ∩ ((0 + 1)...(♯‘𝐹))))
24		elnn0uz 12783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (ℤ_≥‘0))
25	16, 24	sylib 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ (ℤ_≥‘0))
26		fzpreddisj 13479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → ({0} ∩ ((0 + 1)...(♯‘𝐹))) = ∅)
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ ((0 + 1)...(♯‘𝐹))) = ∅)
28	23, 27	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ (1...(♯‘𝐹))) = ∅)
29		undif5 4434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (({0} ∩ (1...(♯‘𝐹))) = ∅ → (({0} ∪ (1...(♯‘𝐹))) ∖ (1...(♯‘𝐹))) = {0})
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ ℕ → (({0} ∪ (1...(♯‘𝐹))) ∖ (1...(♯‘𝐹))) = {0})
31	19, 30	eqtrd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) ∈ ℕ → ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))) = {0})
32	31	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))) = {0})
33	15, 32	eqtrd 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (dom 𝑃 ∖ (1...(♯‘𝐹))) = {0})
34	33	imaeq2d 6014	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) = (𝑃 “ {0}))
35		ffn 6657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
36		0elfz 13530	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐹)))
37	16, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) ∈ ℕ → 0 ∈ (0...(♯‘𝐹)))
38	35, 37	anim12i 613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))))
39	38	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))))
40		fnsnfv 6907	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ 0 ∈ (0...(♯‘𝐹))) → {(𝑃‘0)} = (𝑃 “ {0}))
41	39, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → {(𝑃‘0)} = (𝑃 “ {0}))
42	34, 41	eqtr4d 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) = {(𝑃‘0)})
43		elfz1end 13460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) ∈ ℕ ↔ (♯‘𝐹) ∈ (1...(♯‘𝐹)))
44	43	biimpi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ (1...(♯‘𝐹)))
45	44	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (♯‘𝐹) ∈ (1...(♯‘𝐹)))
46	45	fvresd 6848	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((𝑃 ↾ (1...(♯‘𝐹)))‘(♯‘𝐹)) = (𝑃‘(♯‘𝐹)))
47		ffun 6660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
48	47	funresd 6530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun (𝑃 ↾ (1...(♯‘𝐹))))
49	48	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → Fun (𝑃 ↾ (1...(♯‘𝐹))))
50		ssdmres 5967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1...(♯‘𝐹)) ⊆ dom 𝑃 ↔ dom (𝑃 ↾ (1...(♯‘𝐹))) = (1...(♯‘𝐹)))
51	12, 50	sylib 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → dom (𝑃 ↾ (1...(♯‘𝐹))) = (1...(♯‘𝐹)))
52	45, 51	eleqtrrd 2834	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (♯‘𝐹) ∈ dom (𝑃 ↾ (1...(♯‘𝐹))))
53	49, 52	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (Fun (𝑃 ↾ (1...(♯‘𝐹))) ∧ (♯‘𝐹) ∈ dom (𝑃 ↾ (1...(♯‘𝐹)))))
54		fvelrn 7015	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun (𝑃 ↾ (1...(♯‘𝐹))) ∧ (♯‘𝐹) ∈ dom (𝑃 ↾ (1...(♯‘𝐹)))) → ((𝑃 ↾ (1...(♯‘𝐹)))‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((𝑃 ↾ (1...(♯‘𝐹)))‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))))
56	46, 55	eqeltrrd 2832	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))))
57		eleq1 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ((𝑃‘0) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹)))))
58	57	3ad2ant3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((𝑃‘0) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))) ↔ (𝑃‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹)))))
59	56, 58	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃‘0) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))))
60	59	snssd 4760	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → {(𝑃‘0)} ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))
61	42, 60	eqsstrd 3964	. . . . . . . . . . . . . 14 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))
62	13, 61	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))
63	62	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝐹) ∈ ℕ → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))))
64	63	com3l 89	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))))
65	6, 64	sylbir 235	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝐹)) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))))
66	65	expcom 413	. . . . . . . . 9 ⊢ (1 ≤ (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ₀ → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))))))
67	66	com14 96	. . . . . . . 8 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → ((♯‘𝐹) ∈ ℕ₀ → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (1 ≤ (♯‘𝐹) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))))))
68	4, 5, 67	sylc 65	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (1 ≤ (♯‘𝐹) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))))
69	2, 68	syl 17	. . . . . 6 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (1 ≤ (♯‘𝐹) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))))
70	69	imp 406	. . . . 5 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (1 ≤ (♯‘𝐹) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))))
71	1, 70	sylbi 217	. . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (1 ≤ (♯‘𝐹) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))))
72	71	impcom 407	. . 3 ⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))))
73		imadifssran 6104	. . . . 5 ⊢ ((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) → ((𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))) → ran 𝑃 = ran (𝑃 ↾ (1...(♯‘𝐹)))))
74	73	imp 406	. . . 4 ⊢ (((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))) → ran 𝑃 = ran (𝑃 ↾ (1...(♯‘𝐹))))
75	74	fveq2d 6832	. . 3 ⊢ (((Rel 𝑃 ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) ∧ (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) ⊆ ran (𝑃 ↾ (1...(♯‘𝐹)))) → (♯‘ran 𝑃) = (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))))
76	72, 75	syl 17	. 2 ⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran 𝑃) = (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))))
77		cyclispth 29782	. . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)
78		pthdifv 29715	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺))
79	47	adantl 481	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → Fun 𝑃)
80		fzfid 13886	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (0...(♯‘𝐹)) ∈ Fin)
81		fnfi 9093	. . . . . . . . . . . . . 14 ⊢ ((𝑃 Fn (0...(♯‘𝐹)) ∧ (0...(♯‘𝐹)) ∈ Fin) → 𝑃 ∈ Fin)
82	35, 80, 81	syl2anr 597	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝑃 ∈ Fin)
83		1eluzge0 12784	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (ℤ_≥‘0)
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ (ℤ_≥‘0))
85		fzss1 13469	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
86	84, 85	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1...(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
87	86	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (1...(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
88	10	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → dom 𝑃 = (0...(♯‘𝐹)))
89	87, 88	sseqtrrd 3967	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (1...(♯‘𝐹)) ⊆ dom 𝑃)
90	79, 82, 89	3jca 1128	. . . . . . . . . . . 12 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃))
91	90	ex 412	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → (Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃)))
92	5, 4, 91	sylc 65	. . . . . . . . . 10 ⊢ (𝐹(Walks‘𝐺)𝑃 → (Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃))
93	2, 92	syl 17	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → (Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃))
94	93	adantr 480	. . . . . . . 8 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃))
95		hashres 14351	. . . . . . . 8 ⊢ ((Fun 𝑃 ∧ 𝑃 ∈ Fin ∧ (1...(♯‘𝐹)) ⊆ dom 𝑃) → (♯‘(𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘(1...(♯‘𝐹))))
96	94, 95	syl 17	. . . . . . 7 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (♯‘(𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘(1...(♯‘𝐹))))
97		ovexd 7387	. . . . . . . 8 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (1...(♯‘𝐹)) ∈ V)
98		hashf1rn 14265	. . . . . . . 8 ⊢ (((1...(♯‘𝐹)) ∈ V ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (♯‘(𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))))
99	97, 98	sylancom 588	. . . . . . 7 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (♯‘(𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))))
100	2, 5	syl 17	. . . . . . . . 9 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
101		hashfz1 14259	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘(1...(♯‘𝐹))) = (♯‘𝐹))
102	100, 101	syl 17	. . . . . . . 8 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘(1...(♯‘𝐹))) = (♯‘𝐹))
103	102	adantr 480	. . . . . . 7 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (♯‘(1...(♯‘𝐹))) = (♯‘𝐹))
104	96, 99, 103	3eqtr3d 2774	. . . . . 6 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘𝐹))
105	104	ex 412	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → ((𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺) → (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘𝐹)))
106	78, 105	mpd 15	. . . 4 ⊢ (𝐹(Paths‘𝐺)𝑃 → (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘𝐹))
107	77, 106	syl 17	. . 3 ⊢ (𝐹(Cycles‘𝐺)𝑃 → (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘𝐹))
108	107	adantl 481	. 2 ⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran (𝑃 ↾ (1...(♯‘𝐹)))) = (♯‘𝐹))
109	76, 108	eqtrd 2766	1 ⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran 𝑃) = (♯‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4282 {csn 4575 class class class wbr 5093 dom cdm 5619 ran crn 5620 ↾ cres 5621 “ cima 5622 Rel wrel 5624 Fun wfun 6481 Fn wfn 6482 ⟶wf 6483 –1-1→wf1 6484 ‘cfv 6487 (class class class)co 7352 Fincfn 8875 0cc0 11012 1c1 11013 + caddc 11015 ≤ cle 11153 ℕcn 12131 ℕ₀cn0 12387 ℤ_≥cuz 12738 ...cfz 13413 ♯chash 14243 Vtxcvtx 28981 Walkscwlks 29582 Pathscpths 29695 Cyclesccycls 29770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-dju 9800 df-card 9838 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-n0 12388 df-xnn0 12461 df-z 12475 df-uz 12739 df-fz 13414 df-fzo 13561 df-hash 14244 df-word 14427 df-wlks 29585 df-trls 29676 df-pths 29699 df-cycls 29772

This theorem is referenced by: cycl3grtri 48052

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