Theorem catidd 17754

Description: Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
catidd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
catidd.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
catidd.o	⊢ (𝜑 → · = (comp‘𝐶))
catidd.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catidd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥))
catidd.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓)
catidd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = 𝑓)

Assertion

Ref	Expression
catidd	⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 ))

Distinct variable groups:   𝑦,𝑓, 1   𝑥,𝐵   𝑥,𝑓,𝐶,𝑦   𝜑,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑦, 𝑓)   · (𝑥, 𝑦, 𝑓)   1 (𝑥)   𝐻(𝑥, 𝑦, 𝑓)

Proof of Theorem catidd

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		catidd.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥))) → ( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓)
2	1	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥)) → ( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓))
3		catidd.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
4	3	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐶)))
5	3	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐶)))
6		catidd.h	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
7	6	oveqd 7433	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦𝐻𝑥) = (𝑦(Hom ‘𝐶)𝑥))
8	7	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑓 ∈ (𝑦𝐻𝑥) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)))
9	4, 5, 8	3anbi123d 1464	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑦𝐻𝑥)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))))
10		catidd.o	. . . . . . . . . . . . 13 ⊢ (𝜑 → · = (comp‘𝐶))
11	10	oveqd 7433	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑦, 𝑥⟩ · 𝑥) = (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥))
12	11	oveqd 7433	. . . . . . . . . . 11 ⊢ (𝜑 → ( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
13	12	eqeq1d 2767	. . . . . . . . . 10 ⊢ (𝜑 → (( 1 (⟨𝑦, 𝑥⟩ · 𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
14	2, 9, 13	3imtr3d 296	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
15	14	3expd 1372	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))))
16	15	imp41 431	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
17	16	ralrimiva 3159	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
18		catidd.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦))) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = 𝑓)
19	18	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦)) → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = 𝑓))
20	6	oveqd 7433	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
21	20	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
22	4, 5, 21	3anbi123d 1464	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))))
23	10	oveqd 7433	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨𝑥, 𝑥⟩ · 𝑦) = (⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦))
24	23	oveqd 7433	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
25	24	eqeq1d 2767	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑓(⟨𝑥, 𝑥⟩ · 𝑦) 1 ) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
26	19, 22, 25	3imtr3d 296	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
27	26	3expd 1372	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → (𝑦 ∈ (Base‘𝐶) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))))
28	27	imp41 431	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
29	28	ralrimiva 3159	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)
30	17, 29	jca 521	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
31	30	ralrimiva 3159	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
32		catidd.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ (𝑥𝐻𝑥))
33	32	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 1 ∈ (𝑥𝐻𝑥)))
34	6	oveqd 7433	. . . . . . . 8 ⊢ (𝜑 → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
35	34	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → ( 1 ∈ (𝑥𝐻𝑥) ↔ 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
36	33, 4, 35	3imtr3d 296	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥)))
37	36	imp 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 1 ∈ (𝑥(Hom ‘𝐶)𝑥))
38		eqid 2765	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
39		eqid 2765	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
40		eqid 2765	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
41		catidd.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
42	41	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
43		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
44	38, 39, 40, 42, 43	catideu 17749	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))
45		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑔 = 1 → (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓))
46	45	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑔 = 1 → ((𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
47	46	ralbidv 3190	. . . . . . . 8 ⊢ (𝑔 = 1 → (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ ∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓))
48		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑔 = 1 → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ))
49	48	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑔 = 1 → ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
50	49	ralbidv 3190	. . . . . . . 8 ⊢ (𝑔 = 1 → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓))
51	47, 50	anbi12d 644	. . . . . . 7 ⊢ (𝑔 = 1 → ((∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ (∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)))
52	51	ralbidv 3190	. . . . . 6 ⊢ (𝑔 = 1 → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓)))
53	52	riota2 7398	. . . . 5 ⊢ (( 1 ∈ (𝑥(Hom ‘𝐶)𝑥) ∧ ∃!𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 ))
54	37, 44, 53	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)( 1 (⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦) 1 ) = 𝑓) ↔ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 ))
55	31, 54	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓)) = 1 )
56	55	mpteq2dva 5206	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))) = (𝑥 ∈ (Base‘𝐶) ↦ 1 ))
57		eqid 2765	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
58	38, 39, 40, 41, 57	cidfval 17750	. 2 ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ (Base‘𝐶) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓))))
59	3	mpteq1d 5203	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 1 ) = (𝑥 ∈ (Base‘𝐶) ↦ 1 ))
60	56, 58, 59	3eqtr4d 2810	1 ⊢ (𝜑 → (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ 1 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369  ⟨cop 4597   ↦ cmpt 5194  ‘cfv 6540  ℩crio 7372  (class class class)co 7416  Basecbs 17287  Hom chom 17339  compcco 17340  Catccat 17738  Idccid 17739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-cat 17742  df-cid 17743

This theorem is used by:  iscatd2  17755