Theorem grpoideu 30842

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 30841	. . 3 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)))
3		simpll 778	. . . . . . . . 9 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑧) = 𝑧)
4	3	ralimi 3102	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
5		oveq2 7420	. . . . . . . . . 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 3243	. . . . . . . 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 486	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
11	9	ad2antlr 739	. . . . . . . 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 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
13	12	ralimi 3102	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
14		oveq2 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑦𝐺𝑧) = (𝑦𝐺𝑤))
15	14	eqeq1d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑦𝐺𝑧) = 𝑢 ↔ (𝑦𝐺𝑤) = 𝑢))
16		oveq1 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑧𝐺𝑦) = (𝑤𝐺𝑦))
17	16	eqeq1d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑧𝐺𝑦) = 𝑢 ↔ (𝑤𝐺𝑦) = 𝑢))
18	15, 17	anbi12d 643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
19	18	rexbidv 3189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
20	19	rspcva 3580	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
21	20	adantll 726	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
22	13, 21	sylan2 604	. . . . . . . . . . . . . 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 30840	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
24	22, 23	syldan 602	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
25	24	an32s 664	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
26	25	adantllr 731	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
27	26	adantr 485	. . . . . . . . . 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 7420	. . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
32	31	adantll 726	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
33	32	ad2ant2rl 761	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑤𝐺𝑢) = 𝑢)
34	33	adantllr 731	. . . . . . . . . 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 7420	. . . . . . . . . . . . 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 3580	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) → (𝑢𝐺𝑤) = 𝑤)
39	38	ad2ant2lr 760	. . . . . . . . . 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 417	. . . . . . . 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 707	. . . . . . 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 3157	. . . . . 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 520	. . . . 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 417	. . . 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 3178	. . 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 7419	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢𝐺𝑥) = (𝑤𝐺𝑥))
49	48	eqeq1d 2765	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑤𝐺𝑥) = 𝑥))
50	49	ralbidv 3188	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥))
51	50	reu8 3697	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  ran crn 5664  (class class class)co 7412  GrpOpcgr 30822

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-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  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-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-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7415  df-grpo 30826

This theorem is referenced by:  grpoidval  30846  grpoidcl  30847  grpoidinv2  30848  cnidOLD  30915  hilid  31494