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Theorem cidpropd 16576
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 16562 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
32adantr 466 . . . 4 ((𝜑𝐶 ∈ Cat) → (Base‘𝐶) = (Base‘𝐷))
4 eqid 2770 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2770 . . . . . . . . . 10 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2770 . . . . . . . . . 10 (Hom ‘𝐷) = (Hom ‘𝐷)
71ad4antr 704 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Homf𝐶) = (Homf𝐷))
8 simpr 471 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
9 simpllr 752 . . . . . . . . . 10 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
104, 5, 6, 7, 8, 9homfeqval 16563 . . . . . . . . 9 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐷)𝑥))
11 eqid 2770 . . . . . . . . . . 11 (comp‘𝐶) = (comp‘𝐶)
12 eqid 2770 . . . . . . . . . . 11 (comp‘𝐷) = (comp‘𝐷)
131ad5antr 708 . . . . . . . . . . 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 708 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (compf𝐶) = (compf𝐷))
16 simplr 744 . . . . . . . . . . 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 762 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
18 simpr 471 . . . . . . . . . . 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 752 . . . . . . . . . . 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 16574 . . . . . . . . . 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 2772 . . . . . . . . 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 3302 . . . . . . . 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 16563 . . . . . . . . 9 (((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
247adantr 466 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Homf𝐶) = (Homf𝐷))
2514ad5antr 708 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (compf𝐶) = (compf𝐷))
269adantr 466 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
27 simplr 744 . . . . . . . . . . 11 ((((((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
28 simpllr 752 . . . . . . . . . . 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 471 . . . . . . . . . . 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 16574 . . . . . . . . . 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 2772 . . . . . . . . 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 3302 . . . . . . . 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 608 . . . . . . 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 3133 . . . . . 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 6769 . . . . 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 697 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Homf𝐶) = (Homf𝐷))
37 simpr 471 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
384, 5, 6, 36, 37, 37homfeqval 16563 . . . . . 6 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥(Hom ‘𝐷)𝑥))
392ad2antrr 697 . . . . . . 7 (((𝜑𝐶 ∈ Cat) ∧ 𝑥 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
4039raleqdv 3292 . . . . . 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 6756 . . . . 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 2804 . . . 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 4864 . . 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 471 . . . 4 ((𝜑𝐶 ∈ Cat) → 𝐶 ∈ Cat)
45 eqid 2770 . . . 4 (Id‘𝐶) = (Id‘𝐶)
464, 5, 11, 44, 45cidfval 16543 . . 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 2770 . . . 4 (Base‘𝐷) = (Base‘𝐷)
48 catpropd.3 . . . . . 6 (𝜑𝐶𝑉)
49 catpropd.4 . . . . . 6 (𝜑𝐷𝑊)
501, 14, 48, 49catpropd 16575 . . . . 5 (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
5150biimpa 462 . . . 4 ((𝜑𝐶 ∈ Cat) → 𝐷 ∈ Cat)
52 eqid 2770 . . . 4 (Id‘𝐷) = (Id‘𝐷)
5347, 6, 12, 51, 52cidfval 16543 . . 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 2814 . 2 ((𝜑𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
55 simpr 471 . . . . 5 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ Cat)
56 cidffn 16545 . . . . . . 7 Id Fn Cat
57 fndm 6130 . . . . . . 7 (Id Fn Cat → dom Id = Cat)
5856, 57ax-mp 5 . . . . . 6 dom Id = Cat
5958eleq2i 2841 . . . . 5 (𝐶 ∈ dom Id ↔ 𝐶 ∈ Cat)
6055, 59sylnibr 318 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐶 ∈ dom Id)
61 ndmfv 6359 . . . 4 𝐶 ∈ dom Id → (Id‘𝐶) = ∅)
6260, 61syl 17 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = ∅)
6358eleq2i 2841 . . . . . . 7 (𝐷 ∈ dom Id ↔ 𝐷 ∈ Cat)
6450, 63syl6bbr 278 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ dom Id))
6564notbid 307 . . . . 5 (𝜑 → (¬ 𝐶 ∈ Cat ↔ ¬ 𝐷 ∈ dom Id))
6665biimpa 462 . . . 4 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → ¬ 𝐷 ∈ dom Id)
67 ndmfv 6359 . . . 4 𝐷 ∈ dom Id → (Id‘𝐷) = ∅)
6866, 67syl 17 . . 3 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐷) = ∅)
6962, 68eqtr4d 2807 . 2 ((𝜑 ∧ ¬ 𝐶 ∈ Cat) → (Id‘𝐶) = (Id‘𝐷))
7054, 69pm2.61dan 796 1 (𝜑 → (Id‘𝐶) = (Id‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  c0 4061  cop 4320  cmpt 4861  dom cdm 5249   Fn wfn 6026  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Homf chomf 16533  compfccomf 16534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-cat 16535  df-cid 16536  df-homf 16537  df-comf 16538
This theorem is referenced by:  funcpropd  16766  curfpropd  17080
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