Theorem iccpval

Description: Partition consisting of a fixed number M of parts. (Contributed by AV, 9-Jul-2020)

	Ref	Expression
Assertion	iccpval	⊢ ( 𝑀 ∈ ℕ → ( RePart ‘ 𝑀 ) = { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑚 = 𝑀 → ( 0 ... 𝑚 ) = ( 0 ... 𝑀 ) )
2	1	oveq2d	⊢ ( 𝑚 = 𝑀 → ( ℝ* ↑m ( 0 ... 𝑚 ) ) = ( ℝ* ↑m ( 0 ... 𝑀 ) ) )
3		oveq2	⊢ ( 𝑚 = 𝑀 → ( 0 ..^ 𝑚 ) = ( 0 ..^ 𝑀 ) )
4	3	raleqdv	⊢ ( 𝑚 = 𝑀 → ( ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ↔ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) )
5	2 4	rabeqbidv	⊢ ( 𝑚 = 𝑀 → { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑚 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } = { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } )
6		df-iccp	⊢ RePart = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑚 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } )
7		ovex	⊢ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∈ V
8	7	rabex	⊢ { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } ∈ V
9	5 6 8	fvmpt	⊢ ( 𝑀 ∈ ℕ → ( RePart ‘ 𝑀 ) = { 𝑝 ∈ ( ℝ* ↑m ( 0 ... 𝑀 ) ) ∣ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) } )