Theorem isinv

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	invfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	invfval.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
	invfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	invfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	invfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	invfval.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
Assertion	isinv	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ↔ ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐺 ( 𝑌 𝑆 𝑋 ) 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		invfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		invfval.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
3		invfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		invfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		invfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		invfval.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
7	1 2 3 4 5 6	invfval	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) = ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) )
8	7	breqd	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ↔ 𝐹 ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) 𝐺 ) )
9		brin	⊢ ( 𝐹 ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) 𝐺 ↔ ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐹 ◡ ( 𝑌 𝑆 𝑋 ) 𝐺 ) )
10	8 9	bitrdi	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ↔ ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐹 ◡ ( 𝑌 𝑆 𝑋 ) 𝐺 ) ) )
11		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
12		eqid	⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
13		eqid	⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
14	1 11 12 13 6 3 5 4	sectss	⊢ ( 𝜑 → ( 𝑌 𝑆 𝑋 ) ⊆ ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
15		relxp	⊢ Rel ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
16		relss	⊢ ( ( 𝑌 𝑆 𝑋 ) ⊆ ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) → ( Rel ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) → Rel ( 𝑌 𝑆 𝑋 ) ) )
17	14 15 16	mpisyl	⊢ ( 𝜑 → Rel ( 𝑌 𝑆 𝑋 ) )
18		relbrcnvg	⊢ ( Rel ( 𝑌 𝑆 𝑋 ) → ( 𝐹 ◡ ( 𝑌 𝑆 𝑋 ) 𝐺 ↔ 𝐺 ( 𝑌 𝑆 𝑋 ) 𝐹 ) )
19	17 18	syl	⊢ ( 𝜑 → ( 𝐹 ◡ ( 𝑌 𝑆 𝑋 ) 𝐺 ↔ 𝐺 ( 𝑌 𝑆 𝑋 ) 𝐹 ) )
20	19	anbi2d	⊢ ( 𝜑 → ( ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐹 ◡ ( 𝑌 𝑆 𝑋 ) 𝐺 ) ↔ ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐺 ( 𝑌 𝑆 𝑋 ) 𝐹 ) ) )
21	10 20	bitrd	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ↔ ( 𝐹 ( 𝑋 𝑆 𝑌 ) 𝐺 ∧ 𝐺 ( 𝑌 𝑆 𝑋 ) 𝐹 ) ) )