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Metamath Proof Explorer

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 ) ) } )