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Metamath Proof Explorer

Theorem invfval

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 invfval ( 𝜑 → ( 𝑋 𝑁 𝑌 ) = ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) )

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 6 invffval ⊢ ( 𝜑 → 𝑁 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( ( 𝑥 𝑆 𝑦 ) ∩ ◡ ( 𝑦 𝑆 𝑥 ) ) ) )
8 simprl ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 )
9 simprr ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 )
10 8 9 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 𝑆 𝑦 ) = ( 𝑋 𝑆 𝑌 ) )
11 9 8 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑦 𝑆 𝑥 ) = ( 𝑌 𝑆 𝑋 ) )
12 11 cnveqd ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ◡ ( 𝑦 𝑆 𝑥 ) = ◡ ( 𝑌 𝑆 𝑋 ) )
13 10 12 ineq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑥 𝑆 𝑦 ) ∩ ◡ ( 𝑦 𝑆 𝑥 ) ) = ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) )
14 ovex ⊢ ( 𝑋 𝑆 𝑌 ) ∈ V
15 14 inex1 ⊢ ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) ∈ V
16 15 a1i ⊢ ( 𝜑 → ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) ∈ V )
17 7 13 4 5 16 ovmpod ⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) = ( ( 𝑋 𝑆 𝑌 ) ∩ ◡ ( 𝑌 𝑆 𝑋 ) ) )