Theorem grpoideu 31144

Description: The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoideu	⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)

Distinct variable groups:   𝑥,𝑢,𝐺   𝑢,𝑋,𝑥

Proof of Theorem grpoideu

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpfo.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2	1	grpoidinv 31143	. . 3 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)))
3		simpll 779	. . . . . . . . 9 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑧) = 𝑧)
4	3	ralimi 3103	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
5		oveq2 7454	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
6		id 23	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7	5, 6	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
8	7	cbvralvw 3244	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
9	4, 8	sylib 221	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
10	9	adantl 487	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
11	9	ad2antlr 740	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
12		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
13	12	ralimi 3103	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
14		oveq2 7454	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑦𝐺𝑧) = (𝑦𝐺𝑤))
15	14	eqeq1d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑦𝐺𝑧) = 𝑢 ↔ (𝑦𝐺𝑤) = 𝑢))
16		oveq1 7453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑧𝐺𝑦) = (𝑤𝐺𝑦))
17	16	eqeq1d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑧𝐺𝑦) = 𝑢 ↔ (𝑤𝐺𝑦) = 𝑢))
18	15, 17	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
19	18	rexbidv 3190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
20	19	rspcva 3578	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
21	20	adantll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
22	13, 21	sylan2 605	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
23	1	grpoidinvlem4 31142	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
24	22, 23	syldan 603	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
25	24	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
26	25	adantllr 732	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
27	26	adantr 486	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
28		oveq2 7454	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑤𝐺𝑥) = (𝑤𝐺𝑢))
29		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
30	28, 29	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝑤𝐺𝑥) = 𝑥 ↔ (𝑤𝐺𝑢) = 𝑢))
31	30	rspcva 3578	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
32	31	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
33	32	ad2ant2rl 762	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑤𝐺𝑢) = 𝑢)
34	33	adantllr 732	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑤𝐺𝑢) = 𝑢)
35		oveq2 7454	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑢𝐺𝑥) = (𝑢𝐺𝑤))
36		id 23	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
37	35, 36	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑤) = 𝑤))
38	37	rspcva 3578	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) → (𝑢𝐺𝑤) = 𝑤)
39	38	ad2ant2lr 761	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑢𝐺𝑤) = 𝑤)
40	27, 34, 39	3eqtr3d 2806	. . . . . . . . 9 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → 𝑢 = 𝑤)
41	40	ex 418	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → ((∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → 𝑢 = 𝑤))
42	11, 41	mpand 708	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤))
43	42	ralrimiva 3158	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤))
44	10, 43	jca 521	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤)))
45	44	ex 418	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤))))
46	45	reximdva 3179	. . 3 ⊢ (𝐺 ∈ GrpOp → (∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤))))
47	2, 46	mpd 16	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤)))
48		oveq1 7453	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢𝐺𝑥) = (𝑤𝐺𝑥))
49	48	eqeq1d 2765	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑤𝐺𝑥) = 𝑥))
50	49	ralbidv 3189	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥))
51	50	reu8 3694	. 2 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∃𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤)))
52	47, 51	sylibr 237	1 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367  ran crn 5657  (class class class)co 7446  GrpOpcgr 31124

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7449  df-grpo 31128

This theorem is used by:  grpoidval  31148  grpoidcl  31149  grpoidinv2  31150  cnidOLD  31217  hilid  31796