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Theorem cidpropd 16968
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 → (Homf𝐶) = (Homf𝐷))
catpropd.2 (𝜑 → (compf𝐶) = (compf𝐷))
catpropd.3 (𝜑𝐶𝑉)
catpropd.4 (𝜑𝐷𝑊)
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.
StepHypRef Expression
1 catpropd.1 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
21homfeqbas 16954 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
32adantr 481 . . . 4 ((𝜑𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
4 eqid 2818 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2818 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2818 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
71ad4antr 728 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Homf𝐶) = (Homf𝐷))
8 simpr 485 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
9 simpllr 772 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
104, 5, 6, 7, 8, 9homfeqval 16955 . . . . . . . . 9 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐷)𝑥))
11 eqid 2818 . . . . . . . . . . 11 (comp‘𝐶) = (comp‘𝐶)
12 eqid 2818 . . . . . . . . . . 11 (comp‘𝐷) = (comp‘𝐷)
131ad5antr 730 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (Homf𝐶) = (Homf𝐷))
14 catpropd.2 . . . . . . . . . . . 12 (𝜑 → (compf𝐶) = (compf𝐷))
1514ad5antr 730 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (compf𝐶) = (compf𝐷))
16 simplr 765 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑦 ∈ (Base‘𝐶))
17 simp-4r 780 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
18 simpr 485 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
19 simpllr 772 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
204, 5, 11, 12, 13, 15, 16, 17, 17, 18, 19comfeqval 16966 . . . . . . . . . 10 ((((((𝜑𝐶 ∈ 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))
2120eqeq1d 2820 . . . . . . . . 9 ((((((𝜑𝐶 ∈ 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))
2210, 21raleqbidva 3423 . . . . . . . 8 (((((𝜑𝐶 ∈ 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))
234, 5, 6, 7, 9, 8homfeqval 16955 . . . . . . . . 9 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
247adantr 481 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Homf𝐶) = (Homf𝐷))
2514ad5antr 730 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (compf𝐶) = (compf𝐷))
269adantr 481 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27 simplr 765 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
28 simpllr 772 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
29 simpr 485 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
304, 5, 11, 12, 24, 25, 26, 26, 27, 28, 29comfeqval 16966 . . . . . . . . . 10 ((((((𝜑𝐶 ∈ 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))
3130eqeq1d 2820 . . . . . . . . 9 ((((((𝜑𝐶 ∈ 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))
3223, 31raleqbidva 3423 . . . . . . . 8 (((((𝜑𝐶 ∈ 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))
3322, 32anbi12d 630 . . . . . . 7 (((((𝜑𝐶 ∈ 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)))
3433ralbidva 3193 . . . . . 6 ((((𝜑𝐶 ∈ 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)))
3534riotabidva 7122 . . . . 5 (((𝜑𝐶 ∈ 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)))
361ad2antrr 722 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Homf𝐶) = (Homf𝐷))
37 simpr 485 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
384, 5, 6, 36, 37, 37homfeqval 16955 . . . . . 6 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥(Hom ‘𝐷)𝑥))
392ad2antrr 722 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
4039raleqdv 3413 . . . . . 6 (((𝜑𝐶 ∈ 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)))
4138, 40riotaeqbidv 7106 . . . . 5 (((𝜑𝐶 ∈ 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)))
4235, 41eqtrd 2853 . . . 4 (((𝜑𝐶 ∈ 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)))
433, 42mpteq12dva 5141 . . 3 ((𝜑𝐶 ∈ 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 485 . . . 4 ((𝜑𝐶 ∈ Cat) → 𝐶 ∈ Cat)
45 eqid 2818 . . . 4 (Id‘𝐶) = (Id‘𝐶)
464, 5, 11, 44, 45cidfval 16935 . . 3 ((𝜑𝐶 ∈ 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 2818 . . . 4 (Base‘𝐷) = (Base‘𝐷)
48 catpropd.3 . . . . . 6 (𝜑𝐶𝑉)
49 catpropd.4 . . . . . 6 (𝜑𝐷𝑊)
501, 14, 48, 49catpropd 16967 . . . . 5 (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
5150biimpa 477 . . . 4 ((𝜑𝐶 ∈ Cat) → 𝐷 ∈ Cat)
52 eqid 2818 . . . 4 (Id‘𝐷) = (Id‘𝐷)
5347, 6, 12, 51, 52cidfval 16935 . . 3 ((𝜑𝐶 ∈ 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))))
5443, 46, 533eqtr4d 2863 . 2 ((𝜑𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
55 simpr 485 . . . . 5 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ Cat)
56 cidffn 16937 . . . . . . 7 Id Fn Cat
57 fndm 6448 . . . . . . 7 (Id Fn Cat → dom Id = Cat)
5856, 57ax-mp 5 . . . . . 6 dom Id = Cat
5958eleq2i 2901 . . . . 5 (𝐶 ∈ dom Id ↔ 𝐶 ∈ Cat)
6055, 59sylnibr 330 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ dom Id)
61 ndmfv 6693 . . . 4 𝐶 ∈ dom Id → (Id‘𝐶) = ∅)
6260, 61syl 17 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = ∅)
6358eleq2i 2901 . . . . . . 7 (𝐷 ∈ dom Id ↔ 𝐷 ∈ Cat)
6450, 63syl6bbr 290 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ dom Id))
6564notbid 319 . . . . 5 (𝜑 → (¬ 𝐶 ∈ Cat ↔ ¬ 𝐷 ∈ dom Id))
6665biimpa 477 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐷 ∈ dom Id)
67 ndmfv 6693 . . . 4 𝐷 ∈ dom Id → (Id‘𝐷) = ∅)
6866, 67syl 17 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐷) = ∅)
6962, 68eqtr4d 2856 . 2 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
7054, 69pm2.61dan 809 1 (𝜑 → (Id‘𝐶) = (Id‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  c0 4288  cop 4563  cmpt 5137  dom cdm 5548   Fn wfn 6343  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  Hom chom 16564  compcco 16565  Catccat 16923  Idccid 16924  Homf chomf 16925  compfccomf 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-cat 16927  df-cid 16928  df-homf 16929  df-comf 16930
This theorem is referenced by:  funcpropd  17158  curfpropd  17471
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