Theorem sectid

Description: The identity is a section of itself. (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	sectid	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 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		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
6		eqid	⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
7	1 5 2 3 4	catidcl	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑋 ) )
8	1 5 2 3 4 6 4 7	catlid	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) )
9		eqid	⊢ ( Sect ‘ 𝐶 ) = ( Sect ‘ 𝐶 )
10	1 5 6 2 9 3 4 4 7 7	issect2	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ( ⟨ 𝑋 , 𝑋 ⟩ ( comp ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ 𝑋 ) ) )
11	8 10	mpbird	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ( 𝑋 ( Sect ‘ 𝐶 ) 𝑋 ) ( 𝐼 ‘ 𝑋 ) )