Theorem issect2

Description: Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	issect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	issect.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	issect.o	⊢ · = ( comp ‘ 𝐶 )
	issect.i	⊢ 1 = ( Id ‘ 𝐶 )
	issect.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
	issect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	issect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	issect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	issect.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
	issect.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) )
Assertion	issect2	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		issect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		issect.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		issect.o	⊢ · = ( comp ‘ 𝐶 )
4		issect.i	⊢ 1 = ( Id ‘ 𝐶 )
5		issect.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
6		issect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
7		issect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		issect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9		issect.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
10		issect.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) )
11	9 10	jca	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ) )
12	1 2 3 4 5 6 7 8	issect	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) )
13		df-3an	⊢ ( ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) ↔ ( ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )
14	12 13	bitrdi	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) )
15	11 14	mpbirand	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )