Theorem estrccat

Description: The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020)

	Ref	Expression
Hypothesis	estrccat.c	⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
Assertion	estrccat	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		estrccat.c	⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
2	1	estrccatid	⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ ( Base ‘ 𝑥 ) ) ) ) )
3	2	simpld	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )