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Metamath Proof Explorer

Theorem limsupgval

Description: Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014) (Revised by Mario Carneiro, 7-May-2016)

Ref Expression
Hypothesis limsupval.1 ⊢ 𝐺 = ( 𝑘 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) )
Assertion limsupgval ( 𝑀 ∈ ℝ → ( 𝐺 ‘ 𝑀 ) = sup ( ( ( 𝐹 “ ( 𝑀 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) )

Proof

Step Hyp Ref Expression
1 limsupval.1 ⊢ 𝐺 = ( 𝑘 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) )
2 oveq1 ⊢ ( 𝑘 = 𝑀 → ( 𝑘 [,) +∞ ) = ( 𝑀 [,) +∞ ) )
3 2 imaeq2d ⊢ ( 𝑘 = 𝑀 → ( 𝐹 “ ( 𝑘 [,) +∞ ) ) = ( 𝐹 “ ( 𝑀 [,) +∞ ) ) )
4 3 ineq1d ⊢ ( 𝑘 = 𝑀 → ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) = ( ( 𝐹 “ ( 𝑀 [,) +∞ ) ) ∩ ℝ* ) )
5 4 supeq1d ⊢ ( 𝑘 = 𝑀 → sup ( ( ( 𝐹 “ ( 𝑘 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) = sup ( ( ( 𝐹 “ ( 𝑀 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) )
6 xrltso ⊢ < Or ℝ*
7 6 supex ⊢ sup ( ( ( 𝐹 “ ( 𝑀 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ V
8 5 1 7 fvmpt ⊢ ( 𝑀 ∈ ℝ → ( 𝐺 ‘ 𝑀 ) = sup ( ( ( 𝐹 “ ( 𝑀 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) )