Theorem initoeu1 18179

Description: Initial objects are essentially unique (strong form), i.e. there is a unique isomorphism between two initial objects, see statement in [Lang] p. 58 ("... if P, P' are two universal objects [...] then there exists a unique isomorphism between them.". (Proposed by BJ, 14-Apr-2020.) (Contributed by AV, 14-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
initoeu1.b	⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Assertion

Ref	Expression
initoeu1	⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝜑,𝑓

Proof of Theorem initoeu1

Dummy variables 𝑎 𝑔 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		initoeu1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
2		eqid 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2761	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		initoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	2, 3, 4	isinitoi 18167	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
6	1, 5	mpdan 700	. 2 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
7		initoeu1.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))
8	2, 3, 4	isinitoi 18167	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (InitO‘𝐶)) → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)))
9	7, 8	mpdan 700	. . . 4 ⊢ (𝜑 → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)))
10		oveq2 7426	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴(Hom ‘𝐶)𝑏) = (𝐴(Hom ‘𝐶)𝐵))
11	10	eleq2d 2847	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
12	11	eubidv 2612	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
13	12	rspcv 3573	. . . . . . 7 ⊢ (𝐵 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
14		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
15	4	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
16		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
17		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
18	2, 3, 14, 15, 16, 17	isohom 17944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
19	18	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
20		euex 2603	. . . . . . . . . . . . . . 15 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵))
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
22		oveq2 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝐵(Hom ‘𝐶)𝑎) = (𝐵(Hom ‘𝐶)𝐴))
23	22	eleq2d 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐴 → (𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) ↔ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
24	23	eubidv 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐴 → (∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) ↔ ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
25	24	rspcva 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
26		euex 2603	. . . . . . . . . . . . . . . . 17 ⊢ (∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
27	25, 26	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
28	27	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
29	28	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
30		eqid 2761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
31	15	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
32	16	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
33	17	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
34	4, 1, 7	2initoinv 18178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
35	34	ad4ant134 1193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
36	2, 30, 31, 32, 33, 14, 35	inviso1 17934	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
37	36	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
38	37	eximdv 1950	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
39	38	expcom 419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
40	39	exlimiv 1963	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
41	40	com3l 90	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → (∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
42	41	impd 416	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → ((∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
43	21, 29, 42	syl2and 620	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
44	43	imp 412	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎))) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
45		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎))) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵))
46		euelss 4278	. . . . . . . . . . . 12 ⊢ (((𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵) ∧ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) ∧ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
47	19, 44, 45, 46	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎))) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
48	47	exp42 441	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ (Base‘𝐶) → (𝐴 ∈ (Base‘𝐶) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
49	48	com24 96	. . . . . . . . 9 ⊢ (𝜑 → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → (𝐴 ∈ (Base‘𝐶) → (𝐵 ∈ (Base‘𝐶) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
50	49	com14 97	. . . . . . . 8 ⊢ (𝐵 ∈ (Base‘𝐶) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → (𝐴 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
51	50	expd 421	. . . . . . 7 ⊢ (𝐵 ∈ (Base‘𝐶) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → (𝐴 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
52	13, 51	syldc 49	. . . . . 6 ⊢ (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝐵 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → (𝐴 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
53	52	com15 102	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
54	53	impd 416	. . . 4 ⊢ (𝜑 → ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
55	9, 54	mpd 16	. . 3 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
56	55	impd 416	. 2 ⊢ (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
57	6, 56	mpd 16	1 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2594  ∀wral 3077   ⊆ wss 3899   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  Hom chom 17432  Catccat 17831  Invcinv 17913  Isociso 17914  InitOcinito 18149

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-cat 17835  df-cid 17836  df-sect 17915  df-inv 17916  df-iso 17917  df-inito 18152

This theorem is used by:  initoeu1w  18180