Theorem cidpropd 17733

Description: Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
catpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
catpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
catpropd.3	⊢ (𝜑 → 𝐶 ∈ 𝑉)
catpropd.4	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
cidpropd	⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))

Proof of Theorem cidpropd

Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catpropd.1	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	homfeqbas 17719	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
4		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	1	ad4antr 742	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
8		simpr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
9		simpllr 785	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
10	4, 5, 6, 7, 8, 9	homfeqval 17720	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐷)𝑥))
11		eqid 2761	. . . . . . . . . . 11 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		eqid 2761	. . . . . . . . . . 11 ⊢ (comp‘𝐷) = (comp‘𝐷)
13	1	ad5antr 744	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (Homf ‘𝐶) = (Homf ‘𝐷))
14		catpropd.2	. . . . . . . . . . . 12 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
15	14	ad5antr 744	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (compf‘𝐶) = (compf‘𝐷))
16		simplr 778	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑦 ∈ (Base‘𝐶))
17		simp-4r 793	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
18		simpr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
19		simpllr 785	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
20	4, 5, 11, 12, 13, 15, 16, 17, 17, 18, 19	comfeqval 17731	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓))
21	20	eqeq1d 2763	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑔(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓 ↔ (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓))
22	10, 21	raleqbidva 3325	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 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 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓))
23	4, 5, 6, 7, 9, 8	homfeqval 17720	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
24	7	adantr 484	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Homf ‘𝐶) = (Homf ‘𝐷))
25	14	ad5antr 744	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (compf‘𝐶) = (compf‘𝐷))
26	9	adantr 484	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27		simplr 778	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
28		simpllr 785	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
29		simpr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
30	4, 5, 11, 12, 24, 25, 26, 26, 27, 28, 29	comfeqval 17731	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔))
31	30	eqeq1d 2763	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓))
32	23, 31	raleqbidva 3325	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)𝑔) = 𝑓 ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓))
33	22, 32	anbi12d 641	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 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) ↔ (∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
34	33	ralbidva 3182	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 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) ↔ ∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
35	34	riotabidva 7367	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 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)) = (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
36	1	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
37		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	4, 5, 6, 36, 37, 37	homfeqval 17720	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥(Hom ‘𝐷)𝑥))
39	2	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
40	39	raleqdv 3319	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓) ↔ ∀𝑦 ∈ (Base‘𝐷)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
41	38, 40	riotaeqbidv 7351	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (℩𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)∀𝑦 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)) = (℩𝑔 ∈ (𝑥(Hom ‘𝐷)𝑥)∀𝑦 ∈ (Base‘𝐷)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
42	35, 41	eqtrd 2796	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 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)) = (℩𝑔 ∈ (𝑥(Hom ‘𝐷)𝑥)∀𝑦 ∈ (Base‘𝐷)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓)))
43	3, 42	mpteq12dva 5183	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (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‘𝐷) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐷)𝑥)∀𝑦 ∈ (Base‘𝐷)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓))))
44		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
45		eqid 2761	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
46	4, 5, 11, 44, 45	cidfval 17699	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (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))))
47		eqid 2761	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
48		catpropd.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
49		catpropd.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
50	1, 14, 48, 49	catpropd 17732	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
51	50	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
52		eqid 2761	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
53	47, 6, 12, 51, 52	cidfval 17699	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (Id‘𝐷) = (𝑥 ∈ (Base‘𝐷) ↦ (℩𝑔 ∈ (𝑥(Hom ‘𝐷)𝑥)∀𝑦 ∈ (Base‘𝐷)(∀𝑓 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)(𝑓(⟨𝑥, 𝑥⟩(comp‘𝐷)𝑦)𝑔) = 𝑓))))
54	43, 46, 53	3eqtr4d 2806	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
55		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ Cat)
56		cidffn 17701	. . . . . . 7 ⊢ Id Fn Cat
57	56	fndmi 6620	. . . . . 6 ⊢ dom Id = Cat
58	57	eleq2i 2853	. . . . 5 ⊢ (𝐶 ∈ dom Id ↔ 𝐶 ∈ Cat)
59	55, 58	sylnibr 331	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ dom Id)
60		ndmfv 6894	. . . 4 ⊢ (¬ 𝐶 ∈ dom Id → (Id‘𝐶) = ∅)
61	59, 60	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = ∅)
62	57	eleq2i 2853	. . . . . . 7 ⊢ (𝐷 ∈ dom Id ↔ 𝐷 ∈ Cat)
63	50, 62	bitr4di 291	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ dom Id))
64	63	notbid 320	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 ∈ Cat ↔ ¬ 𝐷 ∈ dom Id))
65	64	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐷 ∈ dom Id)
66		ndmfv 6894	. . . 4 ⊢ (¬ 𝐷 ∈ dom Id → (Id‘𝐷) = ∅)
67	65, 66	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐷) = ∅)
68	61, 67	eqtr4d 2799	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
69	54, 68	pm2.61dan 822	1 ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∅c0 4283  ⟨cop 4585   ↦ cmpt 5178  dom cdm 5643  ‘cfv 6516  ℩crio 7347  (class class class)co 7391  Basecbs 17236  Hom chom 17288  compcco 17289  Catccat 17687  Idccid 17688  Hom_f chomf 17689  comp_fccomf 17690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-cat 17691  df-cid 17692  df-homf 17693  df-comf 17694

This theorem is referenced by:  funcpropd  17926  curfpropd  18256  sectpropdlem  49618