Theorem usgredg2v 16206

Description: In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgredg2v.v	⊢ 𝑉 = (Vtx‘𝐺)
usgredg2v.e	⊢ 𝐸 = (iEdg‘𝐺)
usgredg2v.a	⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
usgredg2v.f	⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))

Assertion

Ref	Expression
usgredg2v	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Distinct variable groups:   𝑥,𝐸,𝑧   𝑧,𝐺   𝑥,𝑁,𝑧   𝑧,𝑉   𝑦,𝐴   𝑦,𝐸,𝑥,𝑧   𝑦,𝐺   𝑦,𝑁   𝑦,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥)   𝑉(𝑥)

Proof of Theorem usgredg2v

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgredg2v.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		usgredg2v.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	1, 2, 3	usgredg2vlem1 16204	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
5	4	ralrimiva 2615	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
6	5	adantr 276	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
7		simpr 110	. . . . . . . 8 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
8		preq1 3767	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
9	8	eqeq2d 2244	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑦) = {𝑢, 𝑁} ↔ (𝐸‘𝑦) = {𝑧, 𝑁}))
10	9	cbvriotavw 6013	. . . . . . . 8 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})
11	8	eqeq2d 2244	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝐸‘𝑤) = {𝑢, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
12	11	cbvriotavw 6013	. . . . . . . 8 ⊢ (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})
13	7, 10, 12	3eqtr4g 2290	. . . . . . 7 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))
14		eqid 2232	. . . . . . 7 ⊢ 𝑁 = 𝑁
15	13, 14	jctir 313	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁))
16	15	orcd 741	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}))))
17		simpl 109	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
18		simpl 109	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → 𝑦 ∈ 𝐴)
19	17, 18	anim12i 338	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
20	1, 2, 3	usgredg2vlem2 16205	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) → ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) → (𝐸‘𝑦) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁}))
21	19, 10, 20	mpisyl 1492	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑦) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁})
22		an3 591	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴))
23	1, 2, 3	usgredg2vlem2 16205	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑤 ∈ 𝐴) → ((℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁}))
24	22, 12, 23	mpisyl 1492	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐸‘𝑤) = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁})
25	21, 24	eqeq12d 2247	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ {(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁}))
26	2	usgrf1 16157	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→ran 𝐸)
27	26	adantr 276	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐸:dom 𝐸–1-1→ran 𝐸)
28		elrabi 2969	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑦 ∈ dom 𝐸)
29	28, 3	eleq2s 2327	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ dom 𝐸)
30		elrabi 2969	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} → 𝑤 ∈ dom 𝐸)
31	30, 3	eleq2s 2327	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → 𝑤 ∈ dom 𝐸)
32	29, 31	anim12i 338	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸))
33		f1fveq 5944	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑤 ∈ dom 𝐸)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
34	27, 32, 33	syl2an 289	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐸‘𝑦) = (𝐸‘𝑤) ↔ 𝑦 = 𝑤))
35		vtxex 16000	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → (Vtx‘𝐺) ∈ V)
36	1, 35	eqeltrid 2319	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝑉 ∈ V)
37		riotaexg 6006	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
38	36, 37	syl 14	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
39	38	adantr 276	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V)
40		simpr 110	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
41		riotaexg 6006	. . . . . . . . . . 11 ⊢ (𝑉 ∈ V → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
42	36, 41	syl 14	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
43	42	adantr 276	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V)
44		preq12bg 3876	. . . . . . . . 9 ⊢ ((((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉) ∧ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉)) → ({(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
45	39, 40, 43, 40, 44	syl22anc 1275	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ({(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
46	45	adantr 276	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ({(℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}), 𝑁} = {(℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
47	25, 34, 46	3bitr3d 218	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑦 = 𝑤 ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
48	47	adantr 276	. . . . 5 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → (𝑦 = 𝑤 ↔ (((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑢 ∈ 𝑉 (𝐸‘𝑦) = {𝑢, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑢 ∈ 𝑉 (𝐸‘𝑤) = {𝑢, 𝑁})))))
49	16, 48	mpbird 167	. . . 4 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁})) → 𝑦 = 𝑤)
50	49	ex 115	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
51	50	ralrimivva 2624	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
52		usgredg2v.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}))
53		fveqeq2 5678	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝐸‘𝑦) = {𝑧, 𝑁} ↔ (𝐸‘𝑤) = {𝑧, 𝑁}))
54	53	riotabidv 6004	. . 3 ⊢ (𝑦 = 𝑤 → (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}))
55	52, 54	f1mpt 5943	. 2 ⊢ (𝐹:𝐴–1-1→𝑉 ↔ (∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (𝐸‘𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
56	6, 51, 55	sylanbrc 417	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2203  ∀wral 2520  {crab 2524  Vcvv 2812  {cpr 3689   ↦ cmpt 4170  dom cdm 4748  ran crn 4749  –1-1→wf1 5348  ‘cfv 5351  ℩crio 6001  Vtxcvtx 15994  iEdgciedg 15995  USGraphcusgr 16136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-umgren 16076  df-usgren 16138

This theorem is referenced by:  usgriedgdomord  16207