Theorem invid

Description: The inverse of the identity is the identity. (Contributed by AV, 8-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	invid	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 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	1 2 3 4	sectid	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )
6		eqid	⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 )
7		eqid	⊢ ( Sect ‘ 𝐶 ) = ( Sect ‘ 𝐶 )
8	1 6 3 4 4 7	isinv	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ∧ ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ) ) )
9	5 5 8	mpbir2and	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Inv ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )