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Theorem initoeu1 16864
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 (𝜑𝐶 ∈ Cat)
initoeu1.a (𝜑𝐴 ∈ (InitO‘𝐶))
initoeu1.b (𝜑𝐵 ∈ (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.
StepHypRef Expression
1 initoeu1.a . . 3 (𝜑𝐴 ∈ (InitO‘𝐶))
2 eqid 2771 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2771 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
4 initoeu1.c . . . 4 (𝜑𝐶 ∈ Cat)
52, 3, 4isinitoi 16856 . . 3 ((𝜑𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
61, 5mpdan 667 . 2 (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
7 initoeu1.b . . . . 5 (𝜑𝐵 ∈ (InitO‘𝐶))
82, 3, 4isinitoi 16856 . . . . 5 ((𝜑𝐵 ∈ (InitO‘𝐶)) → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)))
97, 8mpdan 667 . . . 4 (𝜑 → (𝐵 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)))
10 oveq2 6800 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(Hom ‘𝐶)𝑏) = (𝐴(Hom ‘𝐶)𝐵))
1110eleq2d 2836 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
1211eubidv 2638 . . . . . . . 8 (𝑏 = 𝐵 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
1312rspcv 3456 . . . . . . 7 (𝐵 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
14 eqid 2771 . . . . . . . . . . . . . 14 (Iso‘𝐶) = (Iso‘𝐶)
154adantr 466 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
16 simprr 756 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
17 simprl 754 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
182, 3, 14, 15, 16, 17isohom 16639 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (𝐴(Iso‘𝐶)𝐵) ⊆ (𝐴(Hom ‘𝐶)𝐵))
1918adantr 466 . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . 15 (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵))
2120a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)))
22 oveq2 6800 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → (𝐵(Hom ‘𝐶)𝑎) = (𝐵(Hom ‘𝐶)𝐴))
2322eleq2d 2836 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐴 → (𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) ↔ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
2423eubidv 2638 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝐴 → (∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) ↔ ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
2524rspcva 3458 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
26 euex 2642 . . . . . . . . . . . . . . . . 17 (∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
2725, 26syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎)) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴))
2827ex 397 . . . . . . . . . . . . . . 15 (𝐴 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑎) → ∃𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)))
2928ad2antll 708 . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . . . . 21 (Inv‘𝐶) = (Inv‘𝐶)
3115ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐶 ∈ Cat)
3216ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐴 ∈ (Base‘𝐶))
3317ad2antrr 705 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝐵 ∈ (Base‘𝐶))
344, 1, 72initoinv 16863 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
35343exp 1112 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)))
3635adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴) → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)))
3736imp31 404 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓(𝐴(Inv‘𝐶)𝐵)𝑔)
382, 30, 31, 32, 33, 14, 37inviso1 16629 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) ∧ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵)) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
3938ex 397 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ (Base‘𝐶) ∧ 𝐴 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝐵(Hom ‘𝐶)𝐴)) → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝐵) → 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
4039eximdv 1998 . . . . . . . . . . . . . . . . . 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)))
4140expcom 398 . . . . . . . . . . . . . . . . 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))))
4241exlimiv 2010 . . . . . . . . . . . . . . . 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))))
4342com3l 89 . . . . . . . . . . . . . . 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))))
4443impd 396 . . . . . . . . . . . . . 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)))
4521, 29, 44syl2and 595 . . . . . . . . . . . . 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)))
4645imp 393 . . . . . . . . . . . 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))
47 simprl 754 . . . . . . . . . . . 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))
48 euelss 4062 . . . . . . . . . . . 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))
4919, 46, 47, 48syl3anc 1476 . . . . . . . . . . 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))
5049exp42 422 . . . . . . . . . 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)))))
5150com24 95 . . . . . . . . 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)))))
5251com14 96 . . . . . . . 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)))))
5352expd 400 . . . . . . 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))))))
5413, 53syldc 48 . . . . . 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))))))
5554com15 101 . . . . 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))))))
5655impd 396 . . . 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)))))
579, 56mpd 15 . . 3 (𝜑 → (𝐴 ∈ (Base‘𝐶) → (∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))))
5857impd 396 . 2 (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)) → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)))
596, 58mpd 15 1 (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  ∃!weu 2618  wral 3061  wss 3723   class class class wbr 4786  cfv 6029  (class class class)co 6792  Basecbs 16060  Hom chom 16156  Catccat 16528  Invcinv 16608  Isociso 16609  InitOcinito 16841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7315  df-2nd 7316  df-cat 16532  df-cid 16533  df-sect 16610  df-inv 16611  df-iso 16612  df-inito 16844
This theorem is referenced by:  initoeu1w  16865
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