Theorem sectfval

Description: Value of the section relation. (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	sectfval	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) = { ⟨ 𝑓 , 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	sectffval	⊢ ( 𝜑 → 𝑆 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ) } ) )
10		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 )
11		simprr	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 )
12	10 11	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
13	12	eleq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ↔ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
14	11 10	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑦 𝐻 𝑥 ) = ( 𝑌 𝐻 𝑋 ) )
15	14	eleq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ↔ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) )
16	13 15	anbi12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ↔ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ) )
17	10 11	opeq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑋 , 𝑌 ⟩ )
18	17 10	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑥 ) = ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) )
19	18	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑥 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) )
20	10	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 1 ‘ 𝑥 ) = ( 1 ‘ 𝑋 ) )
21	19 20	eqeq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ↔ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) )
22	16 21	anbi12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( ( 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 ) ) ) )
23	22	opabbidv	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → { ⟨ 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 ) ) } )
24		ovex	⊢ ( 𝑋 𝐻 𝑌 ) ∈ V
25		ovex	⊢ ( 𝑌 𝐻 𝑋 ) ∈ V
26	24 25	xpex	⊢ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ∈ V
27		opabssxp	⊢ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) )
28	26 27	ssexi	⊢ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ∈ V
29	28	a1i	⊢ ( 𝜑 → { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ∈ V )
30	9 23 7 8 29	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } )