Theorem umgr2wlk 26845

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 25999	. . . . . 6 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
2		umgr2wlk.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
3	2	eleq2i 2693	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∈ 𝐸 ↔ {𝐵, 𝐶} ∈ (Edg‘𝐺))
4		eqid 2622	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	4	uhgredgiedgb 26021	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐵, 𝐶} ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)))
6	3, 5	syl5bb 272	. . . . . 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 219	. . . 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 1256	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑖 ∈ dom (iEdg‘𝐺){𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
11	2	eleq2i 2693	. . . . . . 7 ⊢ ({𝐴, 𝐵} ∈ 𝐸 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
12	4	uhgredgiedgb 26021	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → ({𝐴, 𝐵} ∈ (Edg‘𝐺) ↔ ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗)))
13	11, 12	syl5bb 272	. . . . . 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 219	. . . 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 1256	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑗 ∈ dom (iEdg‘𝐺){𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
18		s2cli 13625	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ ∈ Word V
19		s3cli 13626	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
20	18, 19	pm3.2i 471	. . . . . . . . 9 ⊢ (⟨“𝑗𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V)
21		eqid 2622	. . . . . . . . . 10 ⊢ ⟨“𝑗𝑖”⟩ = ⟨“𝑗𝑖”⟩
22		eqid 2622	. . . . . . . . . 10 ⊢ ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩
23		simpl1 1064	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))) → 𝐺 ∈ UMGraph )
24		3simpc 1060	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))
25	24	adantr 481	. . . . . . . . . 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 473	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗))
27	26	eqcomd 2628	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑗) = {𝐴, 𝐵})
28	27	adantl 482	. . . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖))
30	29	eqcomd 2628	. . . . . . . . . . 11 ⊢ (({𝐴, 𝐵} = ((iEdg‘𝐺)‘𝑗) ∧ {𝐵, 𝐶} = ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐺)‘𝑖) = {𝐵, 𝐶})
31	30	adantl 482	. . . . . . . . . 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 26841	. . . . . . . . 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 4658	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → (𝑓(Walks‘𝐺)𝑝 ↔ ⟨“𝑗𝑖”⟩(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩))
34		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑓 = ⟨“𝑗𝑖”⟩ → (#‘𝑓) = (#‘⟨“𝑗𝑖”⟩))
35	34	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑓 = ⟨“𝑗𝑖”⟩ → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑗𝑖”⟩) = 2))
36	35	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((#‘𝑓) = 2 ↔ (#‘⟨“𝑗𝑖”⟩) = 2))
37		fveq1 6190	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
38	37	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐴 = (𝑝‘0) ↔ 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0)))
39		fveq1 6190	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
40	39	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐵 = (𝑝‘1) ↔ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1)))
41		fveq1 6190	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝑝‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
42	41	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → (𝐶 = (𝑝‘2) ↔ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2)))
43	38, 40, 42	3anbi123d 1399	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨“𝐴𝐵𝐶”⟩ → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
44	43	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑓 = ⟨“𝑗𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵𝐶”⟩) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) ↔ (𝐴 = (⟨“𝐴𝐵𝐶”⟩‘0) ∧ 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘1) ∧ 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘2))))
45	33, 36, 44	3anbi123d 1399	. . . . . . . . . 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)))))
46	45	spc2egv 3295	. . . . . . . . 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)))))
47	20, 32, 46	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))))
48	47	exp32 631	. . . . . . 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))))))
49	48	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))))))
50	49	rexlimivw 3029	. . . . 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))))))
51	50	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))))))
52	51	rexlimivw 3029	. . 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))))))
53	52	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))))))
54	10, 17, 53	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200  {cpr 4179   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  0cc0 9936  1c1 9937  2c2 11070  #chash 13117  Word cword 13291  ⟨“cs2 13586  ⟨“cs3 13587  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UMGraph cumgr 25976  Walkscwlks 26492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-wlks 26495

This theorem is referenced by:  umgr2wlkon  26846  umgrwwlks2on  26850