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Theorem cyclnumvtx 29714

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 29705	. . . . 5 ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		pthiswlk 29639	. . . . . . 7 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3		eqid 2734	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkp 29528	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
5		wlkcl 29527	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
6		elnnnn0c 12538	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ ↔ ((♯‘𝐹) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝐹)))
7		frel 6707	. . . . . . . . . . . . . . . 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 13629	. . . . . . . . . . . . . . . . 17 ⊢ (1...(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
10		fdm 6711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → dom 𝑃 = (0...(♯‘𝐹)))
11	9, 10	sseqtrrid 4000	. . . . . . . . . . . . . . . 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 4098	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (dom 𝑃 ∖ (1...(♯‘𝐹))) = ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))))
16		nnnn0 12500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ ℕ₀)
17		fz0sn0fz1 13651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (0...(♯‘𝐹)) = ({0} ∪ (1...(♯‘𝐹))))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) ∈ ℕ → (0...(♯‘𝐹)) = ({0} ∪ (1...(♯‘𝐹))))
19	18	difeq1d 4098	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ ℕ → ((0...(♯‘𝐹)) ∖ (1...(♯‘𝐹))) = (({0} ∪ (1...(♯‘𝐹))) ∖ (1...(♯‘𝐹))))
20		1e0p1 12742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 = (0 + 1)
21	20	oveq1i 7409	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...(♯‘𝐹)) = ((0 + 1)...(♯‘𝐹))
22	21	ineq2i 4190	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({0} ∩ (1...(♯‘𝐹))) = ({0} ∩ ((0 + 1)...(♯‘𝐹)))
23	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ (1...(♯‘𝐹))) = ({0} ∩ ((0 + 1)...(♯‘𝐹))))
24		elnn0uz 12889	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (ℤ_≥‘0))
25	16, 24	sylib 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ ℕ → (♯‘𝐹) ∈ (ℤ_≥‘0))
26		fzpreddisj 13579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘0) → ({0} ∩ ((0 + 1)...(♯‘𝐹))) = ∅)
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ ((0 + 1)...(♯‘𝐹))) = ∅)
28	23, 27	eqtrd 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) ∈ ℕ → ({0} ∩ (1...(♯‘𝐹))) = ∅)
29		undif5 4458	. . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (dom 𝑃 ∖ (1...(♯‘𝐹))) = {0})
34	33	imaeq2d 6044	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) = (𝑃 “ {0}))
35		ffn 6702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → 𝑃 Fn (0...(♯‘𝐹)))
36		0elfz 13630	. . . . . . . . . . . . . . . . . . . 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 6954	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃 “ (dom 𝑃 ∖ (1...(♯‘𝐹)))) = {(𝑃‘0)})
43		elfz1end 13560	. . . . . . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ((𝑃 ↾ (1...(♯‘𝐹)))‘(♯‘𝐹)) = (𝑃‘(♯‘𝐹)))
47		ffun 6705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) → Fun 𝑃)
48	47	funresd 6575	. . . . . . . . . . . . . . . . . . . . 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 5997	. . . . . . . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . . . . . 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 7062	. . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃‘(♯‘𝐹)) ∈ ran (𝑃 ↾ (1...(♯‘𝐹))))
57		eleq1 2821	. . . . . . . . . . . . . . . . . 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 4782	. . . . . . . . . . . . . . 15 ⊢ ((𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ (♯‘𝐹) ∈ ℕ ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → {(𝑃‘0)} ⊆ ran (𝑃 ↾ (1...(♯‘𝐹))))
61	42, 60	eqsstrd 3991	. . . . . . . . . . . . . 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 6137	. . . . 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 6876	. . 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 29711	. . . 4 ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)
78		pthdifv 29644	. . . . 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 13980	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (0...(♯‘𝐹)) ∈ Fin)
81		fnfi 9186	. . . . . . . . . . . . . 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 12900	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (ℤ_≥‘0)
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ (ℤ_≥‘0))
85		fzss1 13569	. . . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . 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 14444	. . . . . . . 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 7434	. . . . . . . 8 ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺)) → (1...(♯‘𝐹)) ∈ V)
98		hashf1rn 14358	. . . . . . . 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 14352	. . . . . . . . 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 2777	. . . . . 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 2769	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 1539 ∈ wcel 2107 Vcvv 3457 ∖ cdif 3921 ∪ cun 3922 ∩ cin 3923 ⊆ wss 3924 ∅c0 4306 {csn 4599 class class class wbr 5116 dom cdm 5651 ran crn 5652 ↾ cres 5653 “ cima 5654 Rel wrel 5656 Fun wfun 6521 Fn wfn 6522 ⟶wf 6523 –1-1→wf1 6524 ‘cfv 6527 (class class class)co 7399 Fincfn 8953 0cc0 11121 1c1 11122 + caddc 11124 ≤ cle 11262 ℕcn 12232 ℕ₀cn0 12493 ℤ_≥cuz 12844 ...cfz 13513 ♯chash 14336 Vtxcvtx 28907 Walkscwlks 29508 Pathscpths 29624 Cyclesccycls 29699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5246 ax-sep 5263 ax-nul 5273 ax-pow 5332 ax-pr 5399 ax-un 7723 ax-cnex 11177 ax-resscn 11178 ax-1cn 11179 ax-icn 11180 ax-addcl 11181 ax-addrcl 11182 ax-mulcl 11183 ax-mulrcl 11184 ax-mulcom 11185 ax-addass 11186 ax-mulass 11187 ax-distr 11188 ax-i2m1 11189 ax-1ne0 11190 ax-1rid 11191 ax-rnegex 11192 ax-rrecex 11193 ax-cnre 11194 ax-pre-lttri 11195 ax-pre-lttrn 11196 ax-pre-ltadd 11197 ax-pre-mulgt0 11198

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 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-int 4920 df-iun 4966 df-br 5117 df-opab 5179 df-mpt 5199 df-tr 5227 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 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-pred 6287 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6480 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7356 df-ov 7402 df-oprab 7403 df-mpo 7404 df-om 7856 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8379 df-rdg 8418 df-1o 8474 df-2o 8475 df-oadd 8478 df-er 8713 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9907 df-card 9945 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11460 df-neg 11461 df-nn 12233 df-2 12295 df-n0 12494 df-xnn0 12567 df-z 12581 df-uz 12845 df-fz 13514 df-fzo 13661 df-hash 14337 df-word 14520 df-wlks 29511 df-trls 29604 df-pths 29628 df-cycls 29701

This theorem is referenced by: cycl3grtri 47859

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