Theorem termoeu1 18035

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 2726	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2726	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		termoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	2, 3, 4	istermoi 18017	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (TermO‘𝐶)) → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵)))
6	1, 5	mpdan 685	. 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 18017	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (TermO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
9	7, 8	mpdan 685	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)))
10		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎(Hom ‘𝐶)𝐵) = (𝐴(Hom ‘𝐶)𝐵))
11	10	eleq2d 2812	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
12	11	eubidv 2575	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
13	12	rspcv 3603	. . . . . . 7 ⊢ (𝐴 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝑎(Hom ‘𝐶)𝐵) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
14		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
15	4	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
16		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
17		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
18	2, 3, 14, 15, 16, 17	isohom 17787	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
19	18	adantr 479	. . . . . . . . . . . 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 2566	. . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = 𝐵 → (𝑏(Hom ‘𝐶)𝐴) = (𝐵(Hom ‘𝐶)𝐴))
23	22	eleq2d 2812	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = 𝐵 → (𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
24	23	eubidv 2575	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐵 → (∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) ↔ ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
25	24	rspcva 3605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
26		euex 2566	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝑏(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
29	28	ad2antll 727	. . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
31	15	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
32	16	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
33	17	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 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 18034	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
35	34	ad4ant134 1171	. . . . . . . . . . . . . . . . . . . . 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 17777	. . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . 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 1913	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 1926	. . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . 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 606	. . . . . . . . . . . . 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 405	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 4321	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 434	. . . . . . . . . 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 414	. . . . . . 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 409	. . . 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 409	. 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 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃!weu 2557  ∀wral 3051   ⊆ wss 3946   class class class wbr 5145  ‘cfv 6546  (class class class)co 7416  Basecbs 17208  Hom chom 17272  Catccat 17672  Invcinv 17756  Isociso 17757  TermOctermo 17999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-cat 17676  df-cid 17677  df-sect 17758  df-inv 17759  df-iso 17760  df-termo 18002

This theorem is referenced by:  termoeu1w  18036