Theorem iscat 17603

Description: The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
iscat.b	⊢ 𝐵 = (Base‘𝐶)
iscat.h	⊢ 𝐻 = (Hom ‘𝐶)
iscat.o	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
iscat	⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))

Distinct variable groups:   𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧, ·   𝐵,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   𝐶,𝑓,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧   𝑓,𝐻,𝑔,𝑘,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,𝑘)

Proof of Theorem iscat

Dummy variables 𝑏 𝑐 ℎ 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6896	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
2		fveq2 6881	. . . 4 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		iscat.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2791	. . 3 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fvexd 6896	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
6		simpl 484	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
7	6	fveq2d 6885	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
8		iscat.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
9	7, 8	eqtr4di 2791	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
10		fvexd 6896	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) ∈ V)
11		simpll 766	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
12	11	fveq2d 6885	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
13		iscat.o	. . . . . 6 ⊢ · = (comp‘𝐶)
14	12, 13	eqtr4di 2791	. . . . 5 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
15		simpllr 775	. . . . . 6 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑏 = 𝐵)
16		simplr 768	. . . . . . . . 9 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ℎ = 𝐻)
17	16	oveqd 7413	. . . . . . . 8 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑥) = (𝑥𝐻𝑥))
18	16	oveqd 7413	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
19		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑜 = · )
20	19	oveqd 7413	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑥⟩𝑜𝑥) = (⟨𝑦, 𝑥⟩ · 𝑥))
21	20	oveqd 7413	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓))
22	21	eqeq1d 2735	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓))
23	18, 22	raleqbidv 3343	. . . . . . . . . 10 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓))
24	16	oveqd 7413	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
25	19	oveqd 7413	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑥⟩𝑜𝑦) = (⟨𝑥, 𝑥⟩ · 𝑦))
26	25	oveqd 7413	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔))
27	26	eqeq1d 2735	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))
28	24, 27	raleqbidv 3343	. . . . . . . . . 10 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓))
29	23, 28	anbi12d 632	. . . . . . . . 9 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
30	15, 29	raleqbidv 3343	. . . . . . . 8 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
31	17, 30	rexeqbidv 3344	. . . . . . 7 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ↔ ∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓)))
32	16	oveqd 7413	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑦ℎ𝑧) = (𝑦𝐻𝑧))
33	19	oveqd 7413	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑦⟩𝑜𝑧) = (⟨𝑥, 𝑦⟩ · 𝑧))
34	33	oveqd 7413	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))
35	16	oveqd 7413	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑥ℎ𝑧) = (𝑥𝐻𝑧))
36	34, 35	eleq12d 2828	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ↔ (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧)))
37	16	oveqd 7413	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑧ℎ𝑤) = (𝑧𝐻𝑤))
38	19	oveqd 7413	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑦⟩𝑜𝑤) = (⟨𝑥, 𝑦⟩ · 𝑤))
39	19	oveqd 7413	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑦, 𝑧⟩𝑜𝑤) = (⟨𝑦, 𝑧⟩ · 𝑤))
40	39	oveqd 7413	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔) = (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔))
41		eqidd 2734	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑓 = 𝑓)
42	38, 40, 41	oveq123d 7417	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓))
43	19	oveqd 7413	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (⟨𝑥, 𝑧⟩𝑜𝑤) = (⟨𝑥, 𝑧⟩ · 𝑤))
44		eqidd 2734	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → 𝑘 = 𝑘)
45	43, 44, 34	oveq123d 7417	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
46	42, 45	eqeq12d 2749	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)) ↔ ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))
47	37, 46	raleqbidv 3343	. . . . . . . . . . . . 13 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)) ↔ ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))
48	15, 47	raleqbidv 3343	. . . . . . . . . . . 12 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)) ↔ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))
49	36, 48	anbi12d 632	. . . . . . . . . . 11 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
50	32, 49	raleqbidv 3343	. . . . . . . . . 10 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
51	24, 50	raleqbidv 3343	. . . . . . . . 9 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
52	15, 51	raleqbidv 3343	. . . . . . . 8 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
53	15, 52	raleqbidv 3343	. . . . . . 7 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))))
54	31, 53	anbi12d 632	. . . . . 6 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → ((∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))) ↔ (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
55	15, 54	raleqbidv 3343	. . . . 5 ⊢ ((((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) ∧ 𝑜 = · ) → (∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
56	10, 14, 55	sbcied2 3822	. . . 4 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ([(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
57	5, 9, 56	sbcied2 3822	. . 3 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ([(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
58	1, 4, 57	sbcied2 3822	. 2 ⊢ (𝑐 = 𝐶 → ([(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓)))) ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))
59		df-cat 17599	. 2 ⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(⟨𝑦, 𝑥⟩𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(⟨𝑥, 𝑥⟩𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(⟨𝑦, 𝑧⟩𝑜𝑤)𝑔)(⟨𝑥, 𝑦⟩𝑜𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩𝑜𝑤)(𝑔(⟨𝑥, 𝑦⟩𝑜𝑧)𝑓))))}
60	58, 59	elab2g 3668	1 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ Cat ↔ ∀𝑥 ∈ 𝐵 (∃𝑔 ∈ (𝑥𝐻𝑥)∀𝑦 ∈ 𝐵 (∀𝑓 ∈ (𝑦𝐻𝑥)(𝑔(⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥𝐻𝑦)(𝑓(⟨𝑥, 𝑥⟩ · 𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧) ∧ ∀𝑤 ∈ 𝐵 ∀𝑘 ∈ (𝑧𝐻𝑤)((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475  [wsbc 3775  ⟨cop 4630  ‘cfv 6535  (class class class)co 7396  Basecbs 17131  Hom chom 17195  compcco 17196  Catccat 17595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5302

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-ov 7399  df-cat 17599

This theorem is referenced by:  iscatd  17604  catidex  17605  catcocl  17616  catass  17617  catpropd  17640