Theorem issect

Description: The property " F is a section of G ". (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 )
Assertion	issect	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ∧ ( 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	1 2 3 4 5 6 7 8	sectfval	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } )
10	9	breqd	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ 𝐹 { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } 𝐺 ) )
11		oveq12	⊢ ( ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) → ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) )
12	11	ancoms	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) )
13	12	eqeq1d	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ↔ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )
14		eqid	⊢ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) }
15	13 14	brab2a	⊢ ( 𝐹 { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } 𝐺 ↔ ( ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )
16		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 ) ) )
17	15 16	bitr4i	⊢ ( 𝐹 { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) )
18	10 17	bitrdi	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ↔ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝐺 ∈ ( 𝑌 𝐻 𝑋 ) ∧ ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝐹 ) = ( 1 ‘ 𝑋 ) ) ) )