catpropd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > catpropd		Structured version Visualization version GIF version

Theorem catpropd 16192

Description: Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017.)

**Hypotheses**
Ref	Expression
catpropd.1	⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
catpropd.2	⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
catpropd.3	⊢ (𝜑 → 𝐶 ∈ 𝑉)
catpropd.4	⊢ (𝜑 → 𝐷 ∈ 𝑊)

**Assertion**
Ref	Expression
catpropd	⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))

Proof of Theorem catpropd Dummy variables 𝑓 𝑔 ℎ 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 472	. . . . . . . . . . 11 ⊢ (((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
2	1	ralimi 2936	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
3	2	ralimi 2936	. . . . . . . . 9 ⊢ (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
4	3	ralimi 2936	. . . . . . . 8 ⊢ (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
5	4	ralimi 2936	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
6	5	adantl 481	. . . . . 6 ⊢ ((∃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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
7	6	ralimi 2936	. . . . 5 ⊢ (∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
8	7	a1i 11	. . . 4 ⊢ (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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
9		simpl 472	. . . . . . . . . . 11 ⊢ (((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
10	9	ralimi 2936	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
11	10	ralimi 2936	. . . . . . . . 9 ⊢ (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
12	11	ralimi 2936	. . . . . . . 8 ⊢ (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
13	12	ralimi 2936	. . . . . . 7 ⊢ (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
14	13	adantl 481	. . . . . 6 ⊢ ((∃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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
15	14	ralimi 2936	. . . . 5 ⊢ (∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
16	15	a1i 11	. . . 4 ⊢ (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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
17		nfra1 2925	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)
18		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)
19		nfra1 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑧∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)
20		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)
21		nfra1 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑔∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)
22		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧)
23		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = ℎ → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
24	23	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = ℎ → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
25	24	cbvralv 3147	. . . . . . . . . . . . . . 15 ⊢ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
26		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔))
27	26	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧)))
28	27	ralbidv 2969	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧)))
29	25, 28	syl5bb 271	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧)))
30	21, 22, 29	cbvral 3143	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧))
31		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑦(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑤))
32		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤))
33	32	oveqd 6566	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) = (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔))
34		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑥(Hom ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑤))
35	33, 34	eleq12d 2682	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
36	31, 35	raleqbidv 3129	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
37	36	ralbidv 2969	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑧)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
38	30, 37	syl5bb 271	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
39	38	cbvralv 3147	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤))
40		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑧))
41		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑤) = (𝑧(Hom ‘𝐶)𝑤))
42		opeq2 4341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
43	42	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤) = (⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤))
44	43	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
45	44	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
46	41, 45	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
47	40, 46	raleqbidv 3129	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
48	47	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑦)∀ℎ ∈ (𝑦(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
49	39, 48	syl5bb 271	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤)))
50	19, 20, 49	cbvral 3143	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤))
51		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
52		opeq1 4340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
53	52	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤) = (⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤))
54	53	oveqd 6566	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
55		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥(Hom ‘𝐶)𝑤) = (𝑦(Hom ‘𝐶)𝑤))
56	54, 55	eleq12d 2682	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
57	56	ralbidv 2969	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
58	51, 57	raleqbidv 3129	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
59	58	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
60		ralcom 3079	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤))
61	59, 60	syl6bb 275	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
62	61	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ (Base‘𝐶)∀𝑤 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑥(Hom ‘𝐶)𝑧)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑥(Hom ‘𝐶)𝑤) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
63	50, 62	syl5bb 271	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)))
64	17, 18, 63	cbvral 3143	. . . . . . 7 ⊢ (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤))
65	64	biimpi 205	. . . . . 6 ⊢ (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤))
66	65	ancri 573	. . . . 5 ⊢ (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
67		r19.26 3046	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (Base‘𝐶)(∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ↔ (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
68		r19.26 3046	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ (Base‘𝐶)(∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ↔ (∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
69		r19.26 3046	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ↔ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)))
70		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (Base‘𝐶) = (Base‘𝐶)
71		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
72		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (comp‘𝐶) = (comp‘𝐶)
73		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (comp‘𝐷) = (comp‘𝐷)
74		catpropd.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
75	74	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
76	75	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
77	76	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
78		catpropd.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
79	78	ad5antr 766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (comp_f‘𝐶) = (comp_f‘𝐷))
80	79	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (comp_f‘𝐶) = (comp_f‘𝐷))
81		simpllr 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
82	81	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑥 ∈ (Base‘𝐶))
83	82	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → 𝑥 ∈ (Base‘𝐶))
84		simp-4r 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ (Base‘𝐶))
85	84	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → 𝑦 ∈ (Base‘𝐶))
86		simpllr 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → 𝑤 ∈ (Base‘𝐶))
87		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
88	87	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
89		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤))
90	70, 71, 72, 73, 77, 80, 83, 85, 86, 88, 89	comfeqval 16191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓))
91		simpllr 795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ (Base‘𝐶))
92	91	ad4antr 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → 𝑧 ∈ (Base‘𝐶))
93		simp-4r 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
94		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))
95	70, 71, 72, 73, 77, 80, 83, 92, 86, 93, 94	comfeqval 16191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
96	90, 95	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
97	96	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) → (((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
98	97	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) → ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
99		ralbi 3050	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
100	98, 99	syl6 34	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
101	100	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) → ∀𝑤 ∈ (Base‘𝐶)(∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
102	101	impancom 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ∀𝑤 ∈ (Base‘𝐶)(∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
103	102	impr 647	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → ∀𝑤 ∈ (Base‘𝐶)(∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
104		ralbi 3050	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑤 ∈ (Base‘𝐶)(∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) → (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
105	103, 104	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))
106	105	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → (((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
107	106	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
108	107	ralimdva 2945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
109	69, 108	syl5bir 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
110	109	expdimp 452	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
111		ralbi 3050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
112	110, 111	syl6 34	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
113	112	an32s 842	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
114	113	ralimdva 2945	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
115		ralbi 3050	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)(∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
116	114, 115	syl6 34	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
117	116	expimpd 627	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → ((∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
118	117	ralimdva 2945	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑧 ∈ (Base‘𝐶)(∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → ∀𝑧 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
119		ralbi 3050	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (Base‘𝐶)(∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
120	118, 119	syl6 34	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑧 ∈ (Base‘𝐶)(∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
121	68, 120	syl5bir 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → ((∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
122	121	ralimdva 2945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → ∀𝑦 ∈ (Base‘𝐶)(∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
123		ralbi 3050	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (Base‘𝐶)(∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
124	122, 123	syl6 34	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)(∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
125	67, 124	syl5bir 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
126	125	imp 444	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
127	126	an4s 865	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))
128	127	anbi2d 736	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) ∧ (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))) → ((∃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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
129	128	expr 641	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ((∃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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))))
130	129	ralimdva 2945	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤)) → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → ∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))))
131	130	expimpd 627	. . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)(ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) ∈ (𝑦(Hom ‘𝐶)𝑤) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧)) → ∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))))
132		ralbi 3050	. . . . 5 ⊢ (∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))) → (∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) ↔ ∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
133	66, 131, 132	syl56 35	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) → (∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) ↔ ∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))))))
134	8, 16, 133	pm5.21ndd 368	. . 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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))) ↔ ∀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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
135	74	homfeqbas 16179	. . . 4 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
136		eqid 2610	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
137		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
138	70, 71, 136, 75, 137, 137	homfeqval 16180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑥) = (𝑥(Hom ‘𝐷)𝑥))
139	135	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) → (Base‘𝐶) = (Base‘𝐷))
140	75	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
141		simpr 476	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
142		simpllr 795	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
143	70, 71, 136, 140, 141, 142	homfeqval 16180	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑥) = (𝑦(Hom ‘𝐷)𝑥))
144	74	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
145	78	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → (comp_f‘𝐶) = (comp_f‘𝐷))
146		simplr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑦 ∈ (Base‘𝐶))
147		simp-4r 803	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
148		simpr 476	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
149		simpllr 795	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
150	70, 71, 72, 73, 144, 145, 146, 147, 147, 148, 149	comfeqval 16191	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 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))
151	150	eqeq1d 2612	. . . . . . . . 9 ⊢ (((((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 ↔ (𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓))
152	143, 151	raleqbidva 3131	. . . . . . . 8 ⊢ ((((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 ∈ (𝑦(Hom ‘𝐷)𝑥)(𝑔(⟨𝑦, 𝑥⟩(comp‘𝐷)𝑥)𝑓) = 𝑓))
153	70, 71, 136, 140, 142, 141	homfeqval 16180	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
154	74	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
155	78	ad4antr 764	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (comp_f‘𝐶) = (comp_f‘𝐷))
156		simp-4r 803	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
157		simplr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
158		simpllr 795	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥))
159		simpr 476	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑔 ∈ (𝑥(Hom ‘𝐶)𝑥)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
160	70, 71, 72, 73, 154, 155, 156, 156, 157, 158, 159	comfeqval 16191	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 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))
161	160	eqeq1d 2612	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 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))
162	153, 161	raleqbidva 3131	. . . . . . . 8 ⊢ ((((𝜑 ∧ 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))
163	152, 162	anbi12d 743	. . . . . . 7 ⊢ ((((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) ↔ (∀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)))
164	139, 163	raleqbidva 3131	. . . . . 6 ⊢ (((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) ↔ ∀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)))
165	138, 164	rexeqbidva 3132	. . . . 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) ↔ ∃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)))
166	135	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
167	166	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
168	75	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
169		simplr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
170	70, 71, 136, 168, 81, 169	homfeqval 16180	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
171		simpr 476	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
172	70, 71, 136, 168, 169, 171	homfeqval 16180	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐷)𝑧))
173	172	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (𝑦(Hom ‘𝐶)𝑧) = (𝑦(Hom ‘𝐷)𝑧))
174		simpr 476	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
175	70, 71, 72, 73, 76, 79, 82, 84, 91, 87, 174	comfeqval 16191	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
176	70, 71, 136, 168, 81, 171	homfeqval 16180	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → (𝑥(Hom ‘𝐶)𝑧) = (𝑥(Hom ‘𝐷)𝑧))
177	176	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (𝑥(Hom ‘𝐶)𝑧) = (𝑥(Hom ‘𝐷)𝑧))
178	175, 177	eleq12d 2682	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ↔ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧)))
179	166	ad4antr 764	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (Base‘𝐶) = (Base‘𝐷))
180	76	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
181		simp-4r 803	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
182		simpr 476	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → 𝑤 ∈ (Base‘𝐶))
183	70, 71, 136, 180, 181, 182	homfeqval 16180	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → (𝑧(Hom ‘𝐶)𝑤) = (𝑧(Hom ‘𝐷)𝑤))
184	168	ad4antr 764	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
185	78	ad7antr 770	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (comp_f‘𝐶) = (comp_f‘𝐷))
186	169	ad4antr 764	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑦 ∈ (Base‘𝐶))
187	171	ad4antr 764	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ (Base‘𝐶))
188		simplr 788	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑤 ∈ (Base‘𝐶))
189		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
190		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))
191	70, 71, 72, 73, 184, 185, 186, 187, 188, 189, 190	comfeqval 16191	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔))
192	191	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓))
193	81	ad4antr 764	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑥 ∈ (Base‘𝐶))
194		simp-4r 803	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
195	70, 71, 72, 73, 184, 185, 193, 186, 187, 194, 189	comfeqval 16191	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
196	195	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))
197	192, 196	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))
198	183, 197	raleqbidva 3131	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) ∧ 𝑤 ∈ (Base‘𝐶)) → (∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))
199	179, 198	raleqbidva 3131	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) ↔ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))
200	178, 199	anbi12d 743	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))))
201	173, 200	raleqbidva 3131	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))))
202	170, 201	raleqbidva 3131	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 ∈ (Base‘𝐶)) → (∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))))
203	167, 202	raleqbidva 3131	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))))
204	166, 203	raleqbidva 3131	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))) ↔ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))))
205	165, 204	anbi12d 743	. . . 4 ⊢ ((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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))))
206	135, 205	raleqbidva 3131	. . 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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))))
207	134, 206	bitrd 267	. 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) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)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) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))))
208		catpropd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
209	70, 71, 72	iscat 16156	. . 3 ⊢ (𝐶 ∈ 𝑉 → (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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
210	208, 209	syl 17	. 2 ⊢ (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)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐶)∀ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))))))
211		catpropd.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
212		eqid 2610	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
213	212, 136, 73	iscat 16156	. . 3 ⊢ (𝐷 ∈ 𝑊 → (𝐷 ∈ Cat ↔ ∀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) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))))
214	211, 213	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ Cat ↔ ∀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) ∧ ∀𝑦 ∈ (Base‘𝐷)∀𝑧 ∈ (Base‘𝐷)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)((𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐷)𝑧) ∧ ∀𝑤 ∈ (Base‘𝐷)∀ℎ ∈ (𝑧(Hom ‘𝐷)𝑤)((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐷)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐷)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))))))
215	207, 210, 214	3bitr4d 299	1 ⊢ (𝜑 → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⟨cop 4131 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 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-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-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-cat 16152 df-homf 16154 df-comf 16155

This theorem is referenced by: cidpropd 16193 resscat 16335 funcpropd 16383

W3C validator