Theorem catideu 17615

Description: Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
catidex.b	⊢ 𝐵 = (Base‘𝐶)
catidex.h	⊢ 𝐻 = (Hom ‘𝐶)
catidex.o	⊢ · = (comp‘𝐶)
catidex.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catidex.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
catideu	⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))

Distinct variable groups:   𝑓,𝑔,𝑦,𝐵   𝐶,𝑓,𝑔,𝑦   𝜑,𝑔   𝑓,𝑋,𝑔,𝑦   𝑓,𝐻,𝑔,𝑦   · ,𝑓,𝑔,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑓)

Proof of Theorem catideu

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		catidex.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		catidex.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3		catidex.o	. . 3 ⊢ · = (comp‘𝐶)
4		catidex.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		catidex.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	1, 2, 3, 4, 5	catidex 17614	. 2 ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
7		oveq1 7412	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑋) = (𝑋𝐻𝑋))
8		opeq1 4872	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → ⟨𝑦, 𝑋⟩ = ⟨𝑋, 𝑋⟩)
9	8	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (⟨𝑦, 𝑋⟩ · 𝑋) = (⟨𝑋, 𝑋⟩ · 𝑋))
10	9	oveqd 7422	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
11	10	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
12	7, 11	raleqbidv 3342	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
13		oveq2 7413	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑋𝐻𝑦) = (𝑋𝐻𝑋))
14		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑋))
15	14	oveqd 7422	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
16	15	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
17	13, 16	raleqbidv 3342	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
18	12, 17	anbi12d 631	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓)))
19	18	rspcv 3608	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓)))
20	5, 19	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓)))
21	20	ralrimivw 3150	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ (𝑋𝐻𝑋)(∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓)))
22		an3 657	. . . . . . 7 ⊢ (((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓))
23		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
24		id 22	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
25	23, 24	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ))
26	25	rspcv 3608	. . . . . . . 8 ⊢ (ℎ ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ))
27		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
28		id 22	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → 𝑓 = 𝑔)
29	27, 28	eqeq12d 2748	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔))
30	29	rspcv 3608	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓 → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔))
31	26, 30	im2anan9r 621	. . . . . . 7 ⊢ ((𝑔 ∈ (𝑋𝐻𝑋) ∧ ℎ ∈ (𝑋𝐻𝑋)) → ((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓) → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ ∧ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔)))
32		eqtr2 2756	. . . . . . . 8 ⊢ (((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ ∧ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔) → ℎ = 𝑔)
33	32	equcomd 2022	. . . . . . 7 ⊢ (((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ ∧ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔) → 𝑔 = ℎ)
34	22, 31, 33	syl56 36	. . . . . 6 ⊢ ((𝑔 ∈ (𝑋𝐻𝑋) ∧ ℎ ∈ (𝑋𝐻𝑋)) → (((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)) → 𝑔 = ℎ))
35	34	rgen2 3197	. . . . 5 ⊢ ∀𝑔 ∈ (𝑋𝐻𝑋)∀ℎ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)) → 𝑔 = ℎ)
36	35	a1i 11	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (𝑋𝐻𝑋)∀ℎ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)) → 𝑔 = ℎ))
37		oveq1 7412	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
38	37	eqeq1d 2734	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
39	38	ralbidv 3177	. . . . . 6 ⊢ (𝑔 = ℎ → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
40		oveq2 7413	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
41	40	eqeq1d 2734	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓))
42	41	ralbidv 3177	. . . . . 6 ⊢ (𝑔 = ℎ → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓))
43	39, 42	anbi12d 631	. . . . 5 ⊢ (𝑔 = ℎ → ((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)))
44	43	rmo4 3725	. . . 4 ⊢ (∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ↔ ∀𝑔 ∈ (𝑋𝐻𝑋)∀ℎ ∈ (𝑋𝐻𝑋)(((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) ∧ (∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓)) → 𝑔 = ℎ))
45	36, 44	sylibr 233	. . 3 ⊢ (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
46		rmoim 3735	. . 3 ⊢ (∀𝑔 ∈ (𝑋𝐻𝑋)(∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓)) → (∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓) → ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
47	21, 45, 46	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
48		reu5 3378	. 2 ⊢ (∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) ↔ (∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓) ∧ ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓)))
49	6, 47, 48	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  ∃!wreu 3374  ∃*wrmo 3375  ⟨cop 4633  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-cat 17608

This theorem is referenced by:  catidd  17620  catidcl  17622  catlid  17623  catrid  17624