Theorem initoeu2 18072

Description: Initial objects are essentially unique, if A is an initial object, then so is every object that is isomorphic to A. Proposition 7.3 (2) in [Adamek] p. 102. (Contributed by AV, 10-Apr-2020.)

Hypotheses

Ref	Expression
initoeu1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
initoeu1.a	⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
initoeu2.i	⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)

Assertion

Ref	Expression
initoeu2	⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Proof of Theorem initoeu2

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

Step	Hyp	Ref	Expression
1		initoeu2.i	. 2 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵)
2		initoeu1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		ciclcl 17858	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
4	2, 3	sylan 591	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
5		cicrcl 17859	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
6	2, 5	sylan 591	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
7	2	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
8		cicsym 17860	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
9	7, 8	sylan 591	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
10		eqid 2769	. . . . . . . . . . 11 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
11		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
13		simprl 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
14	10, 11, 7, 12, 13	cic 17855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐 ‘𝐶)𝐴 ↔ ∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)))
15		initoeu1.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
16		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17	11, 16, 2	isinitoi 18055	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
18	15, 17	mpdan 699	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
19		oveq2 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → (𝐴(Hom ‘𝐶)𝑎) = (𝐴(Hom ‘𝐶)𝑏))
20	19	eleq2d 2855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑏 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
21	20	eubidv 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑏 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
22	21	rspcva 3588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏))
23		nfv 1941	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎℎ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)
24		nfv 1941	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓 ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)
25		eleq1w 2852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = ℎ → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)))
26	23, 24, 25	cbveuw 2640	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
27		euex 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
28	2	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
29		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (Base‘𝐶))
30	29	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
31		simprll 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
32	11, 16, 10, 28, 30, 31	isohom 17832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝐵(Iso‘𝐶)𝐴) ⊆ (𝐵(Hom ‘𝐶)𝐴))
33	32	sselda 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
34		eqid 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (comp‘𝐶) = (comp‘𝐶)
35	28	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐶 ∈ Cat)
36	30	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐵 ∈ (Base‘𝐶))
37	31	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝐴 ∈ (Base‘𝐶))
38		simprr 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝑏 ∈ (Base‘𝐶))
39	38	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝑏 ∈ (Base‘𝐶))
40		simprl 782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
41		simprr 784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
42	11, 16, 34, 35, 36, 37, 39, 40, 41	catcocl 17740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))
43		simp-4l 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → 𝜑)
44		df-3an 1103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ↔ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)))
45	44	biimpri 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
46	45	ad4antlr 745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
47		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
48	47	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
49	41	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
50		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))
51	2, 15, 11, 16, 10, 34	initoeu2lem2 18071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ∧ (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
52	43, 46, 48, 49, 50, 51	syl113anc 1407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) ∧ (ℎ(⟨𝐵, 𝐴⟩(comp‘𝐶)𝑏)𝑘) ∈ (𝐵(Hom ‘𝐶)𝑏)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
53	42, 52	mpdan 699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) ∧ (𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
54	53	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → ((𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴) ∧ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
55	33, 54	mpand 707	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
56	55	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
57	56	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
58	57	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
59	58	com15 102	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
60	59	expd 420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑏 ∈ (Base‘𝐶) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
61	60	com24 96	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
62	61	com12 33	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
63	62	exlimiv 1957	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
64	27, 63	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
65	26, 64	sylbi 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))))
66	65	pm2.43i 53	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → (𝑏 ∈ (Base‘𝐶) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
67	66	com12 33	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ (Base‘𝐶) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
68	67	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
69	22, 68	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
70	69	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (Base‘𝐶) → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝜑 → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
71	70	com15 102	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
72	71	adantld 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))))
73	18, 72	mpd 16	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
74	73	imp 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
75	74	exlimdv 1960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
76	14, 75	sylbid 243	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐 ‘𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
77	76	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → (𝐵( ≃𝑐 ‘𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
78	9, 77	mpd 16	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
79	78	an32s 664	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
80	79	ralrimiv 3162	. . . . 5 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
81	2	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
82		simprr 784	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
83	11, 16, 81, 82	isinito 18052	. . . . 5 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
84	80, 83	mpbird 260	. . . 4 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (InitO‘𝐶))
85	84	ex 417	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (InitO‘𝐶)))
86	4, 6, 85	mp2and 711	. 2 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (InitO‘𝐶))
87	1, 86	mpdan 699	1 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  ∀wral 3085  ⟨cop 4600   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  Hom chom 17320  compcco 17321  Catccat 17719  Isociso 17802   ≃_𝑐 ccic 17851  InitOcinito 18037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-supp 8156  df-cat 17723  df-cid 17724  df-sect 17803  df-inv 17804  df-iso 17805  df-cic 17852  df-inito 18040

This theorem is referenced by:  termoeu2  49900