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Metamath Proof Explorer

Theorem oppcinv

Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses oppcsect.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
oppcsect.o ⊢ 𝑂 = ( oppCat ‘ 𝐶 )
oppcsect.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
oppcsect.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
oppcsect.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
oppcinv.s ⊢ 𝐼 = ( Inv ‘ 𝐶 )
oppcinv.t ⊢ 𝐽 = ( Inv ‘ 𝑂 )
Assertion oppcinv ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )

Proof

Step Hyp Ref Expression
1 oppcsect.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 oppcsect.o ⊢ 𝑂 = ( oppCat ‘ 𝐶 )
3 oppcsect.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
4 oppcsect.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 oppcsect.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 oppcinv.s ⊢ 𝐼 = ( Inv ‘ 𝐶 )
7 oppcinv.t ⊢ 𝐽 = ( Inv ‘ 𝑂 )
8 incom ⊢ ( ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) ∩ ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) ) = ( ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) ∩ ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) )
9 eqid ⊢ ( Sect ‘ 𝐶 ) = ( Sect ‘ 𝐶 )
10 eqid ⊢ ( Sect ‘ 𝑂 ) = ( Sect ‘ 𝑂 )
11 1 2 3 5 4 9 10 oppcsect2 ⊢ ( 𝜑 → ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) = ◡ ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
12 11 cnveqd ⊢ ( 𝜑 → ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) = ◡ ◡ ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
13 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
14 eqid ⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
15 eqid ⊢ ( Id ‘ 𝐶 ) = ( Id ‘ 𝐶 )
16 1 13 14 15 9 3 5 4 sectss ⊢ ( 𝜑 → ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ⊆ ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
17 relxp ⊢ Rel ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
18 relss ⊢ ( ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ⊆ ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) → ( Rel ( ( 𝑌 ( Hom ‘ 𝐶 ) 𝑋 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) → Rel ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ) )
19 16 17 18 mpisyl ⊢ ( 𝜑 → Rel ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
20 dfrel2 ⊢ ( Rel ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ↔ ◡ ◡ ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) = ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
21 19 20 sylib ⊢ ( 𝜑 → ◡ ◡ ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) = ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
22 12 21 eqtrd ⊢ ( 𝜑 → ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) = ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) )
23 1 2 3 4 5 9 10 oppcsect2 ⊢ ( 𝜑 → ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) = ◡ ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) )
24 22 23 ineq12d ⊢ ( 𝜑 → ( ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) ∩ ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) ) = ( ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ∩ ◡ ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ) )
25 8 24 eqtrid ⊢ ( 𝜑 → ( ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) ∩ ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) ) = ( ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ∩ ◡ ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ) )
26 2 1 oppcbas ⊢ 𝐵 = ( Base ‘ 𝑂 )
27 2 oppccat ⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat )
28 3 27 syl ⊢ ( 𝜑 → 𝑂 ∈ Cat )
29 26 7 28 4 5 10 invfval ⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( ( 𝑋 ( Sect ‘ 𝑂 ) 𝑌 ) ∩ ◡ ( 𝑌 ( Sect ‘ 𝑂 ) 𝑋 ) ) )
30 1 6 3 5 4 9 invfval ⊢ ( 𝜑 → ( 𝑌 𝐼 𝑋 ) = ( ( 𝑌 ( Sect ‘ 𝐶 ) 𝑋 ) ∩ ◡ ( 𝑋 ( Sect ‘ 𝐶 ) 𝑌 ) ) )
31 25 29 30 3eqtr4d ⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )