Theorem loop1cycl 33731

Description: A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)

Assertion

Ref	Expression
loop1cycl	⊢ (𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))

Distinct variable groups:   𝐺,𝑝   𝐴,𝑓,𝑝   𝑓,𝐺

Proof of Theorem loop1cycl

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cyclprop 28741	. . . . . . . . . . . . 13 ⊢ (𝑓(Cycles‘𝐺)𝑝 → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 1 → (𝑝‘(♯‘𝑓)) = (𝑝‘1))
3	2	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑓) = 1 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘1)))
4	3	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑓) = 1 → ((𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))) ↔ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1))))
5	4	biimpd 228	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑓) = 1 → ((𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))) → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1))))
6	1, 5	mpan9 507	. . . . . . . . . . . 12 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)))
7		pthiswlk 28675	. . . . . . . . . . . . 13 ⊢ (𝑓(Paths‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
8	7	anim1i 615	. . . . . . . . . . . 12 ⊢ ((𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)) → (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)))
9	6, 8	syl 17	. . . . . . . . . . 11 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)))
10	9	anim1i 615	. . . . . . . . . 10 ⊢ (((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) ∧ (♯‘𝑓) = 1) → ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)) ∧ (♯‘𝑓) = 1))
11	10	anabss3 673	. . . . . . . . 9 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)) ∧ (♯‘𝑓) = 1))
12		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1) ∧ (♯‘𝑓) = 1) ↔ ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)) ∧ (♯‘𝑓) = 1))
13	11, 12	sylibr 233	. . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1) ∧ (♯‘𝑓) = 1))
14		3ancomb 1099	. . . . . . . 8 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1) ∧ (♯‘𝑓) = 1) ↔ (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1)))
15	13, 14	sylib 217	. . . . . . 7 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1)))
16		wlkl1loop 28586	. . . . . . . . . 10 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝑓(Walks‘𝐺)𝑝) ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺))
17	16	expl 458	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) → ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺)))
18		eqid 2736	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
19	18	uhgrfun 28017	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
20	17, 19	syl11 33	. . . . . . . 8 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → (𝐺 ∈ UHGraph → {(𝑝‘0)} ∈ (Edg‘𝐺)))
21	20	3impb 1115	. . . . . . 7 ⊢ ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1)) → (𝐺 ∈ UHGraph → {(𝑝‘0)} ∈ (Edg‘𝐺)))
22	15, 21	syl 17	. . . . . 6 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝐺 ∈ UHGraph → {(𝑝‘0)} ∈ (Edg‘𝐺)))
23	22	3adant3 1132	. . . . 5 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → (𝐺 ∈ UHGraph → {(𝑝‘0)} ∈ (Edg‘𝐺)))
24		sneq 4596	. . . . . . 7 ⊢ ((𝑝‘0) = 𝐴 → {(𝑝‘0)} = {𝐴})
25	24	eleq1d 2822	. . . . . 6 ⊢ ((𝑝‘0) = 𝐴 → ({(𝑝‘0)} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ (Edg‘𝐺)))
26	25	3ad2ant3 1135	. . . . 5 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → ({(𝑝‘0)} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ (Edg‘𝐺)))
27	23, 26	sylibd 238	. . . 4 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → (𝐺 ∈ UHGraph → {𝐴} ∈ (Edg‘𝐺)))
28	27	exlimivv 1935	. . 3 ⊢ (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → (𝐺 ∈ UHGraph → {𝐴} ∈ (Edg‘𝐺)))
29	28	com12 32	. 2 ⊢ (𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → {𝐴} ∈ (Edg‘𝐺)))
30		edgval 28000	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
31	30	eleq2i 2829	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ ran (iEdg‘𝐺))
32		elrnrexdm 7039	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗)))
33		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ ({𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴})
34	33	rexbii 3097	. . . . . . . . . . . . . 14 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴})
35	32, 34	syl6ib 250	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴}))
36	31, 35	biimtrid 241	. . . . . . . . . . . 12 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴}))
37	19, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴}))
38		df-rex 3074	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴} ↔ ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}))
39	37, 38	syl6ib 250	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴})))
40	18	lp1cycl 29096	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}) → ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩)
41	40	3expib 1122	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}) → ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
42	41	eximdv 1920	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → (∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}) → ∃𝑗⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
43	39, 42	syld 47	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
44		s1len 14494	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝑗”⟩) = 1
45	44	ax-gen 1797	. . . . . . . . . 10 ⊢ ∀𝑗(♯‘⟨“𝑗”⟩) = 1
46		19.29r 1877	. . . . . . . . . 10 ⊢ ((∃𝑗⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ ∀𝑗(♯‘⟨“𝑗”⟩) = 1) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
47	45, 46	mpan2 689	. . . . . . . . 9 ⊢ (∃𝑗⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
48	43, 47	syl6 35	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1)))
49	48	imp 407	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
50		uhgredgn0 28079	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
51		eldifsni 4750	. . . . . . . . . . 11 ⊢ ({𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → {𝐴} ≠ ∅)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ≠ ∅)
53		snnzb 4679	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
54	52, 53	sylibr 233	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → 𝐴 ∈ V)
55		s2fv0 14776	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
56	54, 55	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
57	56	alrimiv 1930	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∀𝑗(⟨“𝐴𝐴”⟩‘0) = 𝐴)
58		19.29r 1877	. . . . . . 7 ⊢ ((∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ ∀𝑗(⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑗((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
59	49, 57, 58	syl2anc 584	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
60		df-3an 1089	. . . . . . 7 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
61	60	exbii 1850	. . . . . 6 ⊢ (∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ ∃𝑗((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
62	59, 61	sylibr 233	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
63		s1cli 14493	. . . . . . . 8 ⊢ ⟨“𝑗”⟩ ∈ Word V
64		breq1 5108	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ↔ ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
65		fveqeq2 6851	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((♯‘𝑓) = 1 ↔ (♯‘⟨“𝑗”⟩) = 1))
66	64, 65	3anbi12d 1437	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ (⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
67	66	rspcev 3581	. . . . . . . 8 ⊢ ((⟨“𝑗”⟩ ∈ Word V ∧ (⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)) → ∃𝑓 ∈ Word V(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
68	63, 67	mpan 688	. . . . . . 7 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓 ∈ Word V(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
69		rexex 3079	. . . . . . 7 ⊢ (∃𝑓 ∈ Word V(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
70	68, 69	syl 17	. . . . . 6 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
71	70	exlimiv 1933	. . . . 5 ⊢ (∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
72	62, 71	syl 17	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑓(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
73		s2cli 14769	. . . . . . 7 ⊢ ⟨“𝐴𝐴”⟩ ∈ Word V
74		breq2 5109	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
75		fveq1 6841	. . . . . . . . . 10 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐴”⟩‘0))
76	75	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
77	74, 76	3anbi13d 1438	. . . . . . . 8 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
78	77	rspcev 3581	. . . . . . 7 ⊢ ((⟨“𝐴𝐴”⟩ ∈ Word V ∧ (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)) → ∃𝑝 ∈ Word V(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
79	73, 78	mpan 688	. . . . . 6 ⊢ ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑝 ∈ Word V(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
80		rexex 3079	. . . . . 6 ⊢ (∃𝑝 ∈ Word V(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
81	79, 80	syl 17	. . . . 5 ⊢ ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
82	81	eximi 1837	. . . 4 ⊢ (∃𝑓(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
83	72, 82	syl 17	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
84	83	ex 413	. 2 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴)))
85	29, 84	impbid 211	1 ⊢ (𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  Vcvv 3445   ∖ cdif 3907  ∅c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  dom cdm 5633  ran crn 5634  Fun wfun 6490  ‘cfv 6496  0cc0 11051  1c1 11052  ♯chash 14230  Word cword 14402  ⟨“cs1 14483  ⟨“cs2 14730  Vtxcvtx 27947  iEdgciedg 27948  Edgcedg 27998  UHGraphcuhgr 28007  Walkscwlks 28544  Pathscpths 28660  Cyclesccycls 28733

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-edg 27999  df-uhgr 28009  df-wlks 28547  df-wlkson 28548  df-trls 28640  df-trlson 28641  df-pths 28664  df-pthson 28666  df-cycls 28735

This theorem is referenced by:  acycgrislfgr  33746