Theorem grpoideu 29225

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 29224	. . 3 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)))
3		simpll 765	. . . . . . . . 9 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑧) = 𝑧)
4	3	ralimi 3084	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
5		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
6		id 22	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
7	5, 6	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
8	7	cbvralvw 3223	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
9	4, 8	sylib 217	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
10	9	adantl 483	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
11	9	ad2antlr 725	. . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
13	12	ralimi 3084	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))
14		oveq2 7354	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑦𝐺𝑧) = (𝑦𝐺𝑤))
15	14	eqeq1d 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑦𝐺𝑧) = 𝑢 ↔ (𝑦𝐺𝑤) = 𝑢))
16		oveq1 7353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → (𝑧𝐺𝑦) = (𝑤𝐺𝑦))
17	16	eqeq1d 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → ((𝑧𝐺𝑦) = 𝑢 ↔ (𝑤𝐺𝑦) = 𝑢))
18	15, 17	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
19	18	rexbidv 3173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢) ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)))
20	19	rspcva 3574	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
21	20	adantll 712	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢)) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢))
22	13, 21	sylan2 594	. . . . . . . . . . . . . 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 29223	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑤) = 𝑢 ∧ (𝑤𝐺𝑦) = 𝑢)) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
24	22, 23	syldan 592	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
25	24	an32s 650	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
26	25	adantllr 717	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑧 ∈ 𝑋 (((𝑢𝐺𝑧) = 𝑧 ∧ (𝑧𝐺𝑢) = 𝑧) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑧) = 𝑢 ∧ (𝑧𝐺𝑦) = 𝑢))) ∧ 𝑤 ∈ 𝑋) → (𝑤𝐺𝑢) = (𝑢𝐺𝑤))
27	26	adantr 482	. . . . . . . . . 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 7354	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑤𝐺𝑥) = (𝑤𝐺𝑢))
29		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → 𝑥 = 𝑢)
30	28, 29	eqeq12d 2753	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → ((𝑤𝐺𝑥) = 𝑥 ↔ (𝑤𝐺𝑢) = 𝑢))
31	30	rspcva 3574	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
32	31	adantll 712	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥) → (𝑤𝐺𝑢) = 𝑢)
33	32	ad2ant2rl 747	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ 𝑤 ∈ 𝑋) ∧ (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥)) → (𝑤𝐺𝑢) = 𝑢)
34	33	adantllr 717	. . . . . . . . . 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 7354	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑢𝐺𝑥) = (𝑢𝐺𝑤))
36		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
37	35, 36	eqeq12d 2753	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑤) = 𝑤))
38	37	rspcva 3574	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) → (𝑢𝐺𝑤) = 𝑤)
39	38	ad2ant2lr 746	. . . . . . . . . 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 2785	. . . . . . . . 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 414	. . . . . . . 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 693	. . . . . . 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 3141	. . . . . 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 513	. . . . 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 414	. . . 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 3163	. . 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 15	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ∧ ∀𝑤 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥 → 𝑢 = 𝑤)))
48		oveq1 7353	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢𝐺𝑥) = (𝑤𝐺𝑥))
49	48	eqeq1d 2739	. . . 4 ⊢ (𝑢 = 𝑤 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑤𝐺𝑥) = 𝑥))
50	49	ralbidv 3172	. . 3 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (𝑤𝐺𝑥) = 𝑥))
51	50	reu8 3686	. 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 233	1 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∀wral 3062  ∃wrex 3071  ∃!wreu 3349  ran crn 5628  (class class class)co 7346  GrpOpcgr 29205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pr 5379  ax-un 7659

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3735  df-csb 3851  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-ov 7349  df-grpo 29209

This theorem is referenced by:  grpoidval  29229  grpoidcl  29230  grpoidinv2  29231  cnidOLD  29298  hilid  29877