Theorem loop1cycl 35124

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 29723	. . . . . . . . . . . . 13 ⊢ (𝑓(Cycles‘𝐺)𝑝 → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 1 → (𝑝‘(♯‘𝑓)) = (𝑝‘1))
3	2	eqeq2d 2740	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑓) = 1 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘1)))
4	3	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑓) = 1 → ((𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))) ↔ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1))))
5	4	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑓) = 1 → ((𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))) → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1))))
6	1, 5	mpan9 506	. . . . . . . . . . . 12 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)))
7		pthiswlk 29655	. . . . . . . . . . . . 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 675	. . . . . . . . 9 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → ((𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)) ∧ (♯‘𝑓) = 1))
12		df-3an 1088	. . . . . . . . 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 234	. . . . . . . 8 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1) ∧ (♯‘𝑓) = 1))
14		3ancomb 1098	. . . . . . . 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 218	. . . . . . 7 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1)))
16		wlkl1loop 29566	. . . . . . . . . 10 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝑓(Walks‘𝐺)𝑝) ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺))
17	16	expl 457	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) → ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺)))
18		eqid 2729	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
19	18	uhgrfun 28993	. . . . . . . . 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 1114	. . . . . . 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 4599	. . . . . . 7 ⊢ ((𝑝‘0) = 𝐴 → {(𝑝‘0)} = {𝐴})
25	24	eleq1d 2813	. . . . . 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 239	. . . 4 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → (𝐺 ∈ UHGraph → {𝐴} ∈ (Edg‘𝐺)))
28	27	exlimivv 1932	. . 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 28976	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
31	30	eleq2i 2820	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ ran (iEdg‘𝐺))
32		elrnrexdm 7061	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗)))
33		eqcom 2736	. . . . . . . . . . . . . . 15 ⊢ ({𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴})
34	33	rexbii 3076	. . . . . . . . . . . . . 14 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴})
35	32, 34	imbitrdi 251	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴}))
36	31, 35	biimtrid 242	. . . . . . . . . . . 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 3054	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴} ↔ ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}))
39	37, 38	imbitrdi 251	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴})))
40	18	lp1cycl 30081	. . . . . . . . . . . 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 1917	. . . . . . . . . 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 14571	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝑗”⟩) = 1
45	44	ax-gen 1795	. . . . . . . . . 10 ⊢ ∀𝑗(♯‘⟨“𝑗”⟩) = 1
46		19.29r 1874	. . . . . . . . . 10 ⊢ ((∃𝑗⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ ∀𝑗(♯‘⟨“𝑗”⟩) = 1) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
47	45, 46	mpan2 691	. . . . . . . . 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 406	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
50		uhgredgn0 29055	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
51		eldifsni 4754	. . . . . . . . . . 11 ⊢ ({𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → {𝐴} ≠ ∅)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ≠ ∅)
53		snnzb 4682	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
54	52, 53	sylibr 234	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → 𝐴 ∈ V)
55		s2fv0 14853	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
56	54, 55	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
57	56	alrimiv 1927	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∀𝑗(⟨“𝐴𝐴”⟩‘0) = 𝐴)
58		19.29r 1874	. . . . . . 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 1088	. . . . . . 7 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
61	60	exbii 1848	. . . . . 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 234	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
63		s1cli 14570	. . . . . . . 8 ⊢ ⟨“𝑗”⟩ ∈ Word V
64		breq1 5110	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ↔ ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
65		fveqeq2 6867	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((♯‘𝑓) = 1 ↔ (♯‘⟨“𝑗”⟩) = 1))
66	64, 65	3anbi12d 1439	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ (⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
67	66	rspcev 3588	. . . . . . . 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 690	. . . . . . 7 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑓 ∈ Word V(𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
69		rexex 3059	. . . . . . 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 1930	. . . . 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 14846	. . . . . . 7 ⊢ ⟨“𝐴𝐴”⟩ ∈ Word V
74		breq2 5111	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
75		fveq1 6857	. . . . . . . . . 10 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐴”⟩‘0))
76	75	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
77	74, 76	3anbi13d 1440	. . . . . . . 8 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
78	77	rspcev 3588	. . . . . . 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 690	. . . . . 6 ⊢ ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) → ∃𝑝 ∈ Word V(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴))
80		rexex 3059	. . . . . 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 1835	. . . 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 412	. 2 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴)))
85	29, 84	impbid 212	1 ⊢ (𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911  ∅c0 4296  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  dom cdm 5638  ran crn 5639  Fun wfun 6505  ‘cfv 6511  0cc0 11068  1c1 11069  ♯chash 14295  Word cword 14478  ⟨“cs1 14560  ⟨“cs2 14807  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974  UHGraphcuhgr 28983  Walkscwlks 29524  Pathscpths 29640  Cyclesccycls 29715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-edg 28975  df-uhgr 28985  df-wlks 29527  df-wlkson 29528  df-trls 29620  df-trlson 29621  df-pths 29644  df-pthson 29646  df-cycls 29717

This theorem is referenced by:  acycgrislfgr  35139