Theorem initoeu2 16666

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 16462	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
4	2, 3	sylan 488	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐴 ∈ (Base‘𝐶))
5		cicrcl 16463	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
6	2, 5	sylan 488	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (Base‘𝐶))
7	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
8		cicsym 16464	. . . . . . . . . 10 ⊢ ((𝐶 ∈ Cat ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
9	7, 8	sylan 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵( ≃𝑐 ‘𝐶)𝐴)
10		eqid 2622	. . . . . . . . . . . 12 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
11		eqid 2622	. . . . . . . . . . . 12 ⊢ (Base‘𝐶) = (Base‘𝐶)
12		simprr 796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
13		simprl 794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
14	10, 11, 7, 12, 13	cic 16459	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐 ‘𝐶)𝐴 ↔ ∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)))
15		initoeu1.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶))
16		eqid 2622	. . . . . . . . . . . . . . . . . 18 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
17	11, 16, 2	isinitoi 16653	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝐴 ∈ (InitO‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
18	15, 17	mpdan 702	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)))
19		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝑏 → (𝐴(Hom ‘𝐶)𝑎) = (𝐴(Hom ‘𝐶)𝑏))
20	19	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑏 → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
21	20	eubidv 2490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑏 → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎) ↔ ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)))
22	21	rspcva 3307	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ (Base‘𝐶) ∧ ∀𝑎 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑎)) → ∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏))
23		nfv 1843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏)
24		nfv 1843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑓 ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)
25		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ℎ → (𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏)))
26	23, 24, 25	cbveu 2505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) ↔ ∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
27		euex 2494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃!ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃ℎ ℎ ∈ (𝐴(Hom ‘𝐶)𝑏))
28	2	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
29		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (Base‘𝐶))
30	29	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
31		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐴 ∈ (Base‘𝐶))
32	31	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝐴 ∈ (Base‘𝐶))
33	11, 16, 10, 28, 30, 32	isohom 16436	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → (𝐵(Iso‘𝐶)𝐴) ⊆ (𝐵(Hom ‘𝐶)𝐴))
34	33	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Hom ‘𝐶)𝐴))
35		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (comp‘𝐶) = (comp‘𝐶)
36	28	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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)
37	30	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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))
38	32	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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))
39		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝑏 ∈ (Base‘𝐶))
40	39	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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))
41		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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))
42		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((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))
43	11, 16, 35, 36, 37, 38, 40, 41, 42	catcocl 16346	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((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))
44		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) → 𝜑)
45	44	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((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))) → 𝜑)
46	45	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((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)
47		df-3an 1039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)) ↔ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)))
48	47	biimpri 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶)) → (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶) ∧ 𝑏 ∈ (Base‘𝐶)))
49	48	ad4antlr 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((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)))
50		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝜑 ∧ ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) ∧ 𝑏 ∈ (Base‘𝐶))) ∧ 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴)) → 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴))
51	50	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((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))
52	42	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((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))
53		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((((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))
54	2, 15, 11, 16, 10, 35	initoeu2lem2 16665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((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)))
55	46, 49, 51, 52, 53, 54	syl113anc 1338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((((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)))
56	43, 55	mpdan 702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((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)))
57	56	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) → (∃!𝑓 𝑓 ∈ (𝐴(Hom ‘𝐶)𝑏) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
58	34, 57	mpand 711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((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))))
59	58	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((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)))))
60	59	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((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)))))
61	60	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (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))))))
62	61	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃!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))))))
63	62	expd 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃!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)))))))
64	63	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃!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)))))))
65	64	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (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)))))))
66	65	exlimiv 1858	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃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)))))))
67	27, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃!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)))))))
68	26, 67	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃!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)))))))
69	68	pm2.43i 52	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!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))))))
70	69	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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))))))
71	70	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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))))))
72	22, 71	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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)))))
73	72	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (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))))))
74	73	com15 101	. . . . . . . . . . . . . . . . 17 ⊢ (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))))))
75	74	adantld 483	. . . . . . . . . . . . . . . 16 ⊢ (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))))))
76	18, 75	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))))
77	76	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
78	77	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
79	78	exlimiv 1858	. . . . . . . . . . . 12 ⊢ (∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
80	79	com12 32	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (∃𝑘 𝑘 ∈ (𝐵(Iso‘𝐶)𝐴) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
81	14, 80	sylbid 230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵( ≃𝑐 ‘𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
82	81	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → (𝐵( ≃𝑐 ‘𝐶)𝐴 → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))))
83	9, 82	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
84	83	an32s 846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝑏 ∈ (Base‘𝐶) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
85	84	imp 445	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) ∧ 𝑏 ∈ (Base‘𝐶)) → ∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
86	85	ralrimiva 2966	. . . . 5 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏))
87	2	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
88		simprr 796	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (Base‘𝐶))
89	11, 16, 87, 88	isinito 16650	. . . . 5 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → (𝐵 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ (Base‘𝐶)∃!𝑔 𝑔 ∈ (𝐵(Hom ‘𝐶)𝑏)))
90	86, 89	mpbird 247	. . . 4 ⊢ (((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) ∧ (𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶))) → 𝐵 ∈ (InitO‘𝐶))
91	90	ex 450	. . 3 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → ((𝐴 ∈ (Base‘𝐶) ∧ 𝐵 ∈ (Base‘𝐶)) → 𝐵 ∈ (InitO‘𝐶)))
92	4, 6, 91	mp2and 715	. 2 ⊢ ((𝜑 ∧ 𝐴( ≃𝑐 ‘𝐶)𝐵) → 𝐵 ∈ (InitO‘𝐶))
93	1, 92	mpdan 702	1 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∀wral 2912  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Isociso 16406   ≃_𝑐 ccic 16455  InitOcinito 16638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-supp 7296  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-cic 16456  df-inito 16641

This theorem is referenced by: (None)