Theorem umgr2wlk 30276

Description: In a multigraph, there is a walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)

Hypothesis

Ref	Expression
umgr2wlk.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
umgr2wlk	⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))

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

Allowed substitution hints:   𝐸(𝑓,𝑝)

Proof of Theorem umgr2wlk

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgruhgr 29432	. . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
2		umgr2wlk.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
3	2	eleq2i 2855	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
4		eqid 2763	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgredgiedgb 29454	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
6	3, 5	bitrid 286	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
7	1, 6	syl 18	. . . . 5 ⊢ (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
8	7	biimpd 232	. . . 4 ⊢ (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
9	8	a1d 26	. . 3 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))))
10	9	3imp 1128	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
11	2	eleq2i 2855	. . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
12	4	uhgredgiedgb 29454	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
13	11, 12	bitrid 286	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
14	1, 13	syl 18	. . . . 5 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
15	14	biimpd 232	. . . 4 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
16	15	a1dd 51	. . 3 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))))
17	16	3imp 1128	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
18		s2cli 14919	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ ∈ Word V
19		s3cli 14920	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
20	18, 19	pm3.2i 475	. . . . . . . . 9 ⊢ (⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V)
21		eqid 2763	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ = ⟨“𝑗𝑖”⟩
22		eqid 2763	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
23		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → 𝐺 ∈ UMGraph)
24		3simpc 1168	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
25	24	adantr 485	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
26		simpl 487	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
27	26	eqcomd 2769	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
28	27	adantl 486	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
29		simpr 489	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
30	29	eqcomd 2769	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
31	30	adantl 486	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
32	2, 4, 21, 22, 23, 25, 28, 31	umgr2adedgwlk 30272	. . . . . . . . 9 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → (⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
33		breq12 5115	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(Walks‘𝐺)𝑝 ↔ ⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
34		fveqeq2 6892	. . . . . . . . . . . 12 ⊢ (𝑓 = ⟨“𝑗𝑖”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
35	34	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
36		fveq1 6882	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
37	36	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐴 = (𝑝‘0) ↔ 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0)))
38		fveq1 6882	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
39	38	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐵 = (𝑝‘1) ↔ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1)))
40		fveq1 6882	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
41	40	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐶 = (𝑝‘2) ↔ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))
42	37, 39, 41	3anbi123d 1464	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
43	42	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
44	33, 35, 43	3anbi123d 1464	. . . . . . . . . 10 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))))
45	44	spc2egv 3559	. . . . . . . . 9 ⊢ ((⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V) → ((⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘⟨“𝑗𝑖”⟩) = 2 ∧ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
46	20, 32, 45	mpsyl 69	. . . . . . . 8 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
47	46	exp32 425	. . . . . . 7 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
48	47	com12 33	. . . . . 6 ⊢ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
49	48	rexlimivw 3162	. . . . 5 ⊢ (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
50	49	com13 89	. . . 4 ⊢ ({𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
51	50	rexlimivw 3162	. . 3 ⊢ (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
52	51	com12 33	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → (∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖) → (∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
53	10, 17, 52	mp2d 50	1 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  {cpr 4592   class class class wbr 5110  dom cdm 5663  ‘cfv 6538  0cc0 11101  1c1 11102  2c2 12296  ♯chash 14368  Word cword 14552  ⟨“cs2 14880  ⟨“cs3 14881  iEdgciedg 29325  Edgcedg 29375  UHGraphcuhgr 29384  UMGraphcumgr 29409  Walkscwlks 29924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-edg 29376  df-uhgr 29386  df-upgr 29410  df-umgr 29411  df-wlks 29927

This theorem is referenced by:  umgr2wlkon  30277  usgrwwlks2on  30285  umgrwwlks2on  30286