Theorem termoeu1 18051

Description: Terminal objects are essentially unique (strong form), i.e. there is a unique isomorphism between two terminal 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, 18-Apr-2020.)

Hypotheses

Ref	Expression
termoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
termoeu1.a	⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
termoeu1.b	⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))

Assertion

Ref	Expression
termoeu1	⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

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

Proof of Theorem termoeu1

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

Step	Hyp	Ref	Expression
1		termoeu1.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶))
2		eqid 2762	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2762	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		termoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	2, 3, 4	istermoi 18033	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵)))
6	1, 5	mpdan 697	. 2 ⊢ (𝜑 → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵)))
7		termoeu1.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶))
8	2, 3, 4	istermoi 18033	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
9	7, 8	mpdan 697	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
10		oveq1 7403	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎(Hom ‘𝐶)𝐵) = (𝐴(Hom ‘𝐶)𝐵))
11	10	eleq2d 2848	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
12	11	eubidv 2613	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
13	12	rspcv 3577	. . . . . . 7 ⊢ (𝐴 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
14		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
15	4	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
16		simprl 780	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
17		simprr 782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
18	2, 3, 14, 15, 16, 17	isohom 17809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
19	18	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
20		euex 2604	. . . . . . . . . . . . . . 15 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵))
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
22		oveq1 7403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝐵 → (𝑏(Hom ‘𝐶)𝐴) = (𝐵(Hom ‘𝐶)𝐴))
23	22	eleq2d 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝐵 → (𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
24	23	eubidv 2613	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐵 → (∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
25	24	rspcva 3579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
26		euex 2604	. . . . . . . . . . . . . . . . 17 ⊢ (∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
27	25, 26	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
28	27	ex 416	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
29	28	ad2antll 739	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
30		eqid 2762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
31	15	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
32	16	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
33	17	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
34	4, 7, 1	2termoinv 18050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
35	34	ad4ant134 1188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (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 17799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
37	36	ex 416	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
38	37	eximdv 1937	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
39	38	expcom 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
40	39	exlimiv 1950	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
41	40	com3l 89	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → (∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
42	41	impd 414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (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 617	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
44	43	imp 410	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴))) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
45		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴))) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵))
46		euelss 4284	. . . . . . . . . . . 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 1390	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴))) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
48	47	exp42 439	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) → (𝐵 ∈ (Base‘𝐶) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
49	48	com24 95	. . . . . . . . 9 ⊢ (𝜑 → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → (𝐵 ∈ (Base‘𝐶) → (𝐴 ∈ (Base‘𝐶) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
50	49	com14 96	. . . . . . . 8 ⊢ (𝐴 ∈ (Base‘𝐶) → ((∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → (𝐵 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
51	50	expd 419	. . . . . . 7 ⊢ (𝐴 ∈ (Base‘𝐶) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → (𝐵 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
52	13, 51	syldc 48	. . . . . 6 ⊢ (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → (𝐵 ∈ (Base‘𝐶) → (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
53	52	com15 101	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → (𝐵 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))))
54	53	impd 414	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → (𝐵 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))))
55	9, 54	mpd 15	. . 3 ⊢ (𝜑 → (𝐵 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
56	55	impd 414	. 2 ⊢ (𝜑 → ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
57	6, 56	mpd 15	1 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∃!weu 2595  ∀wral 3076   ⊆ wss 3904   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Hom chom 17297  Catccat 17696  Invcinv 17778  Isociso 17779  TermOctermo 18015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-cat 17700  df-cid 17701  df-sect 17780  df-inv 17781  df-iso 17782  df-termo 18018

This theorem is referenced by:  termoeu1w  18052  termcterm2  50132  termccisoeu  50135