Theorem cicfval

Description: The set of isomorphic objects of the category c . (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Assertion	cicfval	⊢ ( 𝐶 ∈ Cat → ( ≃𝑐 ‘ 𝐶 ) = ( ( Iso ‘ 𝐶 ) supp ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		df-cic	⊢ ≃𝑐 = ( 𝑐 ∈ Cat ↦ ( ( Iso ‘ 𝑐 ) supp ∅ ) )
2		fveq2	⊢ ( 𝑐 = 𝐶 → ( Iso ‘ 𝑐 ) = ( Iso ‘ 𝐶 ) )
3	2	oveq1d	⊢ ( 𝑐 = 𝐶 → ( ( Iso ‘ 𝑐 ) supp ∅ ) = ( ( Iso ‘ 𝐶 ) supp ∅ ) )
4		id	⊢ ( 𝐶 ∈ Cat → 𝐶 ∈ Cat )
5		ovexd	⊢ ( 𝐶 ∈ Cat → ( ( Iso ‘ 𝐶 ) supp ∅ ) ∈ V )
6	1 3 4 5	fvmptd3	⊢ ( 𝐶 ∈ Cat → ( ≃𝑐 ‘ 𝐶 ) = ( ( Iso ‘ 𝐶 ) supp ∅ ) )