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Theorem loop1cycl 34116

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 29039	. . . . . . . . . . . . 13 ⊢ (𝑓(Cycles‘𝐺)𝑝 → (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		fveq2 6888	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 1 → (𝑝‘(♯‘𝑓)) = (𝑝‘1))
3	2	eqeq2d 2743	. . . . . . . . . . . . . . 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 28973	. . . . . . . . . . . . 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 28884	. . . . . . . . . 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 2732	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
19	18	uhgrfun 28315	. . . . . . . . 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 4637	. . . . . . 7 ⊢ ((𝑝‘0) = 𝐴 → {(𝑝‘0)} = {𝐴})
25	24	eleq1d 2818	. . . . . 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 28298	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
31	30	eleq2i 2825	. . . . . . . . . . . . 13 ⊢ ({𝐴} ∈ (Edg‘𝐺) ↔ {𝐴} ∈ ran (iEdg‘𝐺))
32		elrnrexdm 7087	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({𝐴} ∈ ran (iEdg‘𝐺) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴} = ((iEdg‘𝐺)‘𝑗)))
33		eqcom 2739	. . . . . . . . . . . . . . 15 ⊢ ({𝐴} = ((iEdg‘𝐺)‘𝑗) ↔ ((iEdg‘𝐺)‘𝑗) = {𝐴})
34	33	rexbii 3094	. . . . . . . . . . . . . 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 3071	. . . . . . . . . . 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 29394	. . . . . . . . . . . 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 14552	. . . . . . . . . . 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 28377	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
51		eldifsni 4792	. . . . . . . . . . 11 ⊢ ({𝐴} ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → {𝐴} ≠ ∅)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → {𝐴} ≠ ∅)
53		snnzb 4721	. . . . . . . . . 10 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
54	52, 53	sylibr 233	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴} ∈ (Edg‘𝐺)) → 𝐴 ∈ V)
55		s2fv0 14834	. . . . . . . . 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 14551	. . . . . . . 8 ⊢ ⟨“𝑗”⟩ ∈ Word V
64		breq1 5150	. . . . . . . . . 10 ⊢ (𝑓 = ⟨“𝑗”⟩ → (𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩ ↔ ⟨“𝑗”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
65		fveqeq2 6897	. . . . . . . . . 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 3612	. . . . . . . 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 3076	. . . . . . 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 14827	. . . . . . 7 ⊢ ⟨“𝐴𝐴”⟩ ∈ Word V
74		breq2 5151	. . . . . . . . 9 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑓(Cycles‘𝐺)𝑝 ↔ 𝑓(Cycles‘𝐺)⟨“𝐴𝐴”⟩))
75		fveq1 6887	. . . . . . . . . 10 ⊢ (𝑝 = ⟨“𝐴𝐴”⟩ → (𝑝‘0) = (⟨“𝐴𝐴”⟩‘0))
76	75	eqeq1d 2734	. . . . . . . . 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 3612	. . . . . . 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 3076	. . . . . 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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 ∅c0 4321 𝒫 cpw 4601 {csn 4627 class class class wbr 5147 dom cdm 5675 ran crn 5676 Fun wfun 6534 ‘cfv 6540 0cc0 11106 1c1 11107 ♯chash 14286 Word cword 14460 ⟨“cs1 14541 ⟨“cs2 14788 Vtxcvtx 28245 iEdgciedg 28246 Edgcedg 28296 UHGraphcuhgr 28305 Walkscwlks 28842 Pathscpths 28958 Cyclesccycls 29031

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-edg 28297 df-uhgr 28307 df-wlks 28845 df-wlkson 28846 df-trls 28938 df-trlson 28939 df-pths 28962 df-pthson 28964 df-cycls 29033

This theorem is referenced by: acycgrislfgr 34131

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