Theorem umgr2wlk 29747

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 28904	. . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
2		umgr2wlk.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
3	2	eleq2i 2820	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
4		eqid 2727	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgredgiedgb 28926	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
6	3, 5	bitrid 283	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
7	1, 6	syl 17	. . . . 5 ⊢ (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
8	7	biimpd 228	. . . 4 ⊢ (𝐺 ∈ UMGraph → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
9	8	a1d 25	. . 3 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))))
10	9	3imp 1109	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
11	2	eleq2i 2820	. . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
12	4	uhgredgiedgb 28926	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
13	11, 12	bitrid 283	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
14	1, 13	syl 17	. . . . 5 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
15	14	biimpd 228	. . . 4 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
16	15	a1dd 50	. . 3 ⊢ (𝐺 ∈ UMGraph → ({𝐴, 𝐵} ∈ 𝐸 → ({𝐵, 𝐶} ∈ 𝐸 → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))))
17	16	3imp 1109	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
18		s2cli 14855	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ ∈ Word V
19		s3cli 14856	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
20	18, 19	pm3.2i 470	. . . . . . . . 9 ⊢ (⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V)
21		eqid 2727	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ = ⟨“𝑗𝑖”⟩
22		eqid 2727	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
23		simpl1 1189	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → 𝐺 ∈ UMGraph)
24		3simpc 1148	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
25	24	adantr 480	. . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
27	26	eqcomd 2733	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
28	27	adantl 481	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
30	29	eqcomd 2733	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
31	30	adantl 481	. . . . . . . . . 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 29743	. . . . . . . . 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 5147	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(Walks‘𝐺)𝑝 ↔ ⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
34		fveqeq2 6900	. . . . . . . . . . . 12 ⊢ (𝑓 = ⟨“𝑗𝑖”⟩ → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
35	34	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((♯‘𝑓) = 2 ↔ (♯‘⟨“𝑗𝑖”⟩) = 2))
36		fveq1 6890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
37	36	eqeq2d 2738	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐴 = (𝑝‘0) ↔ 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0)))
38		fveq1 6890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
39	38	eqeq2d 2738	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐵 = (𝑝‘1) ↔ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1)))
40		fveq1 6890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
41	40	eqeq2d 2738	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐶 = (𝑝‘2) ↔ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))
42	37, 39, 41	3anbi123d 1433	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
43	42	adantl 481	. . . . . . . . . . 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 1433	. . . . . . . . . 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 3584	. . . . . . . . 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 68	. . . . . . . 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 420	. . . . . . 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 32	. . . . . 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 3146	. . . . 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 88	. . . 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 3146	. . 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 32	. 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 49	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃wrex 3065  Vcvv 3469  {cpr 4626   class class class wbr 5142  dom cdm 5672  ‘cfv 6542  0cc0 11130  1c1 11131  2c2 12289  ♯chash 14313  Word cword 14488  ⟨“cs2 14816  ⟨“cs3 14817  iEdgciedg 28797  Edgcedg 28847  UHGraphcuhgr 28856  UMGraphcumgr 28881  Walkscwlks 29397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-umgr 28883  df-wlks 29400

This theorem is referenced by:  umgr2wlkon  29748  umgrwwlks2on  29755