Theorem idinv

Description: The inverse of the identity is the identity. Example 3.13 of Adamek p. 28. (Contributed by AV, 9-Apr-2020)

	Ref	Expression
Hypotheses	invid.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	invid.i	⊢ 𝐼 = ( Id ‘ 𝐶 )
	invid.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	invid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	idinv	⊢ ( 𝜑 → ( ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		invid.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		invid.i	⊢ 𝐼 = ( Id ‘ 𝐶 )
3		invid.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		invid.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		eqid	⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 )
6	1 5 3 4 4	invfun	⊢ ( 𝜑 → Fun ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) )
7	1 2 3 4	invid	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )
8		funbrfv	⊢ ( Fun ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) → ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) → ( ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) ) )
9	6 7 8	sylc	⊢ ( 𝜑 → ( ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) )