Theorem cidpropd 16193

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 16179	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
4		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2610	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2610	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	1	ad4antr 764	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
8		simpr 476	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
9		simpllr 795	. . . . . . . . . 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 16180	. . . . . . . . 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 2610	. . . . . . . . . . 11 ⊢ (comp‘𝐶) = (comp‘𝐶)
12		eqid 2610	. . . . . . . . . . 11 ⊢ (comp‘𝐷) = (comp‘𝐷)
13	1	ad5antr 766	. . . . . . . . . . 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 766	. . . . . . . . . . 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 788	. . . . . . . . . . 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 803	. . . . . . . . . . 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 476	. . . . . . . . . . 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 795	. . . . . . . . . . 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 16191	. . . . . . . . . 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 2612	. . . . . . . . 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 3131	. . . . . . . 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 16180	. . . . . . . . 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 480	. . . . . . . . . . 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 766	. . . . . . . . . . 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 480	. . . . . . . . . . 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 788	. . . . . . . . . . 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 795	. . . . . . . . . . 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 476	. . . . . . . . . . 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 16191	. . . . . . . . . 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 2612	. . . . . . . . 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 3131	. . . . . . . 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 743	. . . . . . 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 2968	. . . . . 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 6527	. . . . 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 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Homf ‘𝐶) = (Homf ‘𝐷))
37		simpr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
38	4, 5, 6, 36, 37, 37	homfeqval 16180	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥(Hom ‘𝐷)𝑥))
39	2	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
40	39	raleqdv 3121	. . . . . 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 6514	. . . . 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 2644	. . . 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 4662	. . 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 476	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐶 ∈ Cat)
45		eqid 2610	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
46	4, 5, 11, 44, 45	cidfval 16160	. . 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 2610	. . . 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 16192	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
51	50	biimpa 500	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → 𝐷 ∈ Cat)
52		eqid 2610	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
53	47, 6, 12, 51, 52	cidfval 16160	. . 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 2654	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
55		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ Cat)
56		cidffn 16162	. . . . . . 7 ⊢ Id Fn Cat
57		fndm 5904	. . . . . . 7 ⊢ (Id Fn Cat → dom Id = Cat)
58	56, 57	ax-mp 5	. . . . . 6 ⊢ dom Id = Cat
59	58	eleq2i 2680	. . . . 5 ⊢ (𝐶 ∈ dom Id ↔ 𝐶 ∈ Cat)
60	55, 59	sylnibr 318	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ dom Id)
61		ndmfv 6128	. . . 4 ⊢ (¬ 𝐶 ∈ dom Id → (Id‘𝐶) = ∅)
62	60, 61	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = ∅)
63	58	eleq2i 2680	. . . . . . 7 ⊢ (𝐷 ∈ dom Id ↔ 𝐷 ∈ Cat)
64	50, 63	syl6bbr 277	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ dom Id))
65	64	notbid 307	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 ∈ Cat ↔ ¬ 𝐷 ∈ dom Id))
66	65	biimpa 500	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐷 ∈ dom Id)
67		ndmfv 6128	. . . 4 ⊢ (¬ 𝐷 ∈ dom Id → (Id‘𝐷) = ∅)
68	66, 67	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐷) = ∅)
69	62, 68	eqtr4d 2647	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
70	54, 69	pm2.61dan 828	1 ⊢ (𝜑 → (Id‘𝐶) = (Id‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874  ⟨cop 4131   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Hom_f chomf 16150  comp_fccomf 16151

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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-homf 16154  df-comf 16155

This theorem is referenced by:  funcpropd  16383  curfpropd  16696