Theorem catideu 16159

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 16158	. 2 ⊢ (𝜑 → ∃𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
7		oveq1 6556	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑋) = (𝑋𝐻𝑋))
8		opeq1 4340	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → ⟨𝑦, 𝑋⟩ = ⟨𝑋, 𝑋⟩)
9	8	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (⟨𝑦, 𝑋⟩ · 𝑋) = (⟨𝑋, 𝑋⟩ · 𝑋))
10	9	oveqd 6566	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
11	10	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
12	7, 11	raleqbidv 3129	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
13		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑋𝐻𝑦) = (𝑋𝐻𝑋))
14		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (⟨𝑋, 𝑋⟩ · 𝑦) = (⟨𝑋, 𝑋⟩ · 𝑋))
15	14	oveqd 6566	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
16	15	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
17	13, 16	raleqbidv 3129	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
18	12, 17	anbi12d 743	. . . . . 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 3278	. . . . 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 2950	. . 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		id 22	. . . . . . . 8 ⊢ ((∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨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	22	ad2ant2rl 781	. . . . . . 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))
24		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
25		id 22	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
26	24, 25	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ))
27	26	rspcv 3278	. . . . . . . 8 ⊢ (ℎ ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ))
28		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
29		id 22	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → 𝑓 = 𝑔)
30	28, 29	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓 ↔ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔))
31	30	rspcv 3278	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑋𝐻𝑋) → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓 → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔))
32	27, 31	im2anan9r 877	. . . . . . 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)))
33		eqtr2 2630	. . . . . . . 8 ⊢ (((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ ∧ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔) → ℎ = 𝑔)
34	33	eqcomd 2616	. . . . . . 7 ⊢ (((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = ℎ ∧ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑔) → 𝑔 = ℎ)
35	23, 32, 34	syl56 35	. . . . . 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))
36	35	rgen2a 2960	. . . . 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)
37	36	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))
38		oveq1 6556	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
39	38	eqeq1d 2612	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ (ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
40	39	ralbidv 2969	. . . . . 6 ⊢ (𝑔 = ℎ → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(ℎ(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓))
41		oveq2 6557	. . . . . . . 8 ⊢ (𝑔 = ℎ → (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ))
42	41	eqeq1d 2612	. . . . . . 7 ⊢ (𝑔 = ℎ → ((𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓 ↔ (𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓))
43	42	ralbidv 2969	. . . . . 6 ⊢ (𝑔 = ℎ → (∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)ℎ) = 𝑓))
44	40, 43	anbi12d 743	. . . . 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)))
45	44	rmo4 3366	. . . 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))
46	37, 45	sylibr 223	. . 3 ⊢ (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)(∀𝑓 ∈ (𝑋𝐻𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑋)(𝑓(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑓))
47		rmoim 3374	. . 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)))
48	21, 46, 47	sylc 63	. 2 ⊢ (𝜑 → ∃*𝑔 ∈ (𝑋𝐻𝑋)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑋)(𝑔(⟨𝑦, 𝑋⟩ · 𝑋)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑋𝐻𝑦)(𝑓(⟨𝑋, 𝑋⟩ · 𝑦)𝑔) = 𝑓))
49		reu5 3136	. 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)))
50	6, 48, 49	sylanbrc 695	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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cat 16152

This theorem is referenced by:  catidd  16164  catidcl  16166  catlid  16167  catrid  16168