Theorem termoeu1 18031

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 2735	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		termoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	2, 3, 4	istermoi 18013	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵)))
6	1, 5	mpdan 687	. 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 18013	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
9	7, 8	mpdan 687	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
10		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎(Hom ‘𝐶)𝐵) = (𝐴(Hom ‘𝐶)𝐵))
11	10	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
12	11	eubidv 2585	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
13	12	rspcv 3597	. . . . . . 7 ⊢ (𝐴 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
14		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
15	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
16		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
17		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
18	2, 3, 14, 15, 16, 17	isohom 17789	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
19	18	adantr 480	. . . . . . . . . . . 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 2576	. . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝐵 → (𝑏(Hom ‘𝐶)𝐴) = (𝐵(Hom ‘𝐶)𝐴))
23	22	eleq2d 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝐵 → (𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
24	23	eubidv 2585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐵 → (∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
25	24	rspcva 3599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
26		euex 2576	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
29	28	ad2antll 729	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
31	15	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
32	16	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
33	17	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 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 18030	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
35	34	ad4ant134 1175	. . . . . . . . . . . . . . . . . . . . 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 17779	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . 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 1930	. . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . 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 608	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 770	. . . . . . . . . . . 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 4307	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 435	. . . . . . . . . 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 415	. . . . . . 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 410	. . . 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 410	. 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 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  ∀wral 3051   ⊆ wss 3926   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  Hom chom 17282  Catccat 17676  Invcinv 17758  Isociso 17759  TermOctermo 17995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-cat 17680  df-cid 17681  df-sect 17760  df-inv 17761  df-iso 17762  df-termo 17998

This theorem is referenced by:  termoeu1w  18032  termcterm2  49399  termccisoeu  49402