Theorem initoeu2 17967

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 17747	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
4	2, 3	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
5		cicrcl 17748	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
6	2, 5	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
7	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
8		cicsym 17749	. . . . . . . . 9 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
9	7, 8	sylan 579	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
10		eqid 2724	. . . . . . . . . . 11 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
11		eqid 2724	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		simprr 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
13		simprl 768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
14	10, 11, 7, 12, 13	cic 17744	. . . . . . . . . 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 2724	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17	11, 16, 2	isinitoi 17950	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
18	15, 17	mpdan 684	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
19		oveq2 7409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑏 → (𝐴(Hom ‘𝐶)𝑎) = (𝐴(Hom ‘𝐶)𝑏))
20	19	eleq2d 2811	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑏 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
21	20	eubidv 2572	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑏 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
22	21	rspcva 3602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏))
23		nfv 1909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎℎ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)
24		nfv 1909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓 ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)
25		eleq1w 2808	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = ℎ → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)))
26	23, 24, 25	cbveuw 2593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
27		euex 2563	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
28	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
29		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (Base‘𝐶))
30	29	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
31		simprll 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
32	11, 16, 10, 28, 30, 31	isohom 17721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝐵(Iso‘𝐶)𝐴) ⊆ (𝐵(Hom ‘𝐶)𝐴))
33	32	sselda 3974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
34		eqid 2724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (comp‘𝐶) = (comp‘𝐶)
35	28	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝑏 ∈ (Base‘𝐶))
39	38	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1086	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ↔ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)))
45	44	biimpri 227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
46	45	ad4antlr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
48	47	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 17966	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 101	. . . . . . . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . . . . 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 1925	. . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . 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 52	. . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 101	. . . . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 15	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 1928	. . . . . . . . . 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 239	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐 ‘𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
77	76	adantr 480	. . . . . . . 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 15	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
79	78	an32s 649	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
80	79	ralrimiv 3137	. . . . 5 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
81	2	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
82		simprr 770	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
83	11, 16, 81, 82	isinito 17947	. . . . 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 257	. . . 4 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (InitO‘𝐶))
85	84	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (InitO‘𝐶)))
86	4, 6, 85	mp2and 696	. 2 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (InitO‘𝐶))
87	1, 86	mpdan 684	1 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084  ∃wex 1773   ∈ wcel 2098  ∃!weu 2554  ∀wral 3053  ⟨cop 4626   class class class wbr 5138  ‘cfv 6533  (class class class)co 7401  Basecbs 17142  Hom chom 17206  compcco 17207  Catccat 17606  Isociso 17691   ≃_𝑐 ccic 17740  InitOcinito 17932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-supp 8141  df-cat 17610  df-cid 17611  df-sect 17692  df-inv 17693  df-iso 17694  df-cic 17741  df-inito 17935

This theorem is referenced by: (None)