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Theorem loop1cycl 34804

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 29651	. . . . . . . . . . . . 13 ⊢ (𝑓(Cycles‘𝐺)𝑝 → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 1 → (𝑝‘(♯‘𝑓)) = (𝑝‘1))
3	2	eqeq2d 2736	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑓) = 1 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘1)))
4	3	anbi2d 628	. . . . . . . . . . . . . 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 505	. . . . . . . . . . . 12 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1) → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘1)))
7		pthiswlk 29585	. . . . . . . . . . . . 13 ⊢ (𝑓(Paths‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
8	7	anim1i 613	. . . . . . . . . . . 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 613	. . . . . . . . . 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 1086	. . . . . . . . 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 1096	. . . . . . . 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 29496	. . . . . . . . . 10 ⊢ (((Fun (iEdg‘𝐺) ∧ 𝑓(Walks‘𝐺)𝑝) ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺))
17	16	expl 456	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) → ((𝑓(Walks‘𝐺)𝑝 ∧ ((♯‘𝑓) = 1 ∧ (𝑝‘0) = (𝑝‘1))) → {(𝑝‘0)} ∈ (Edg‘𝐺)))
18		eqid 2725	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
19	18	uhgrfun 28923	. . . . . . . . 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 1112	. . . . . . 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 1129	. . . . 5 ⊢ ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) → (𝐺 ∈ UHGraph → {(𝑝‘0)} ∈ (Edg‘𝐺)))
24		sneq 4634	. . . . . . 7 ⊢ ((𝑝‘0) = 𝐴 → {(𝑝‘0)} = {𝐴})
25	24	eleq1d 2810	. . . . . 6 ⊢ ((𝑝‘0) = 𝐴 → ({(𝑝‘0)} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ (Edg‘𝐺)))
26	25	3ad2ant3 1132	. . . . 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 1927	. . 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 28906	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
31	30	eleq2i 2817	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ ran (iEdg‘𝐺))
32		elrnrexdm 7094	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗)))
33		eqcom 2732	. . . . . . . . . . . . . . 15 ⊢ ({𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴})
34	33	rexbii 3084	. . . . . . . . . . . . . 14 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴})
35	32, 34	imbitrdi 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 3061	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑗) = {𝐴} ↔ ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}))
39	37, 38	imbitrdi 250	. . . . . . . . . 10 ⊢ (𝐺 ∈ UHGraph → ({𝐴} ∈ (Edg‘𝐺) → ∃𝑗(𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴})))
40	18	lp1cycl 30006	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}) → ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩)
41	40	3expib 1119	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UHGraph → ((𝑗 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑗) = {𝐴}) → ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
42	41	eximdv 1912	. . . . . . . . . 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 14588	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝑗”⟩) = 1
45	44	ax-gen 1789	. . . . . . . . . 10 ⊢ ∀𝑗(♯‘⟨“𝑗”⟩) = 1
46		19.29r 1869	. . . . . . . . . 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 405	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗(⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1))
50		uhgredgn0 28985	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
51		eldifsni 4789	. . . . . . . . . . 11 ⊢ ({𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → {𝐴} ≠ ∅)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ≠ ∅)
53		snnzb 4718	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
54	52, 53	sylibr 233	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → 𝐴 ∈ V)
55		s2fv0 14870	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
56	54, 55	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → (⟨“𝐴𝐴”⟩‘0) = 𝐴)
57	56	alrimiv 1922	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∀𝑗(⟨“𝐴𝐴”⟩‘0) = 𝐴)
58		19.29r 1869	. . . . . . 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 582	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → ∃𝑗((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
60		df-3an 1086	. . . . . . 7 ⊢ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ ((⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1) ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
61	60	exbii 1842	. . . . . 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 14587	. . . . . . . 8 ⊢ ⟨“𝑗”⟩ ∈ Word V
64		breq1 5146	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ↔ ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
65		fveqeq2 6901	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((♯‘𝑓) = 1 ↔ (♯‘⟨“𝑗”⟩) = 1))
66	64, 65	3anbi12d 1433	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑗”⟩ → ((𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴) ↔ (⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘⟨“𝑗”⟩) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
67	66	rspcev 3601	. . . . . . . 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 3066	. . . . . . 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 1925	. . . . 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 14863	. . . . . . 7 ⊢ ⟨“𝐴𝐴”⟩ ∈ Word V
74		breq2 5147	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
75		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐴”⟩‘0))
76	75	eqeq1d 2727	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑝‘0) = 𝐴 ↔ (⟨“𝐴𝐴”⟩‘0) = 𝐴))
77	74, 76	3anbi13d 1434	. . . . . . . 8 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ∧ (♯‘𝑓) = 1 ∧ (⟨“𝐴𝐴”⟩‘0) = 𝐴)))
78	77	rspcev 3601	. . . . . . 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 3066	. . . . . 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 1829	. . . 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 411	. 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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 Vcvv 3463 ∖ cdif 3936 ∅c0 4318 𝒫 cpw 4598 {csn 4624 class class class wbr 5143 dom cdm 5672 ran crn 5673 Fun wfun 6537 ‘cfv 6543 0cc0 11138 1c1 11139 ♯chash 14321 Word cword 14496 ⟨“cs1 14577 ⟨“cs2 14824 Vtxcvtx 28853 iEdgciedg 28854 Edgcedg 28904 UHGraphcuhgr 28913 Walkscwlks 29454 Pathscpths 29570 Cyclesccycls 29643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-edg 28905 df-uhgr 28915 df-wlks 29457 df-wlkson 29458 df-trls 29550 df-trlson 29551 df-pths 29574 df-pthson 29576 df-cycls 29645

This theorem is referenced by: acycgrislfgr 34819

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