Theorem ceilval2

Description: The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018)

	Ref	Expression
Assertion	ceilval2	⊢ ( 𝐴 ∈ ℝ → ( ⌈ ‘ 𝐴 ) = ( ℩ 𝑦 ∈ ℤ ( 𝐴 ≤ 𝑦 ∧ 𝑦 < ( 𝐴 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ≤ 𝑦 ↔ 𝐴 ≤ 𝑦 ) )
2		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 + 1 ) = ( 𝐴 + 1 ) )
3	2	breq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑦 < ( 𝑥 + 1 ) ↔ 𝑦 < ( 𝐴 + 1 ) ) )
4	1 3	anbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ≤ 𝑦 ∧ 𝑦 < ( 𝑥 + 1 ) ) ↔ ( 𝐴 ≤ 𝑦 ∧ 𝑦 < ( 𝐴 + 1 ) ) ) )
5	4	riotabidv	⊢ ( 𝑥 = 𝐴 → ( ℩ 𝑦 ∈ ℤ ( 𝑥 ≤ 𝑦 ∧ 𝑦 < ( 𝑥 + 1 ) ) ) = ( ℩ 𝑦 ∈ ℤ ( 𝐴 ≤ 𝑦 ∧ 𝑦 < ( 𝐴 + 1 ) ) ) )
6		dfceil2	⊢ ⌈ = ( 𝑥 ∈ ℝ ↦ ( ℩ 𝑦 ∈ ℤ ( 𝑥 ≤ 𝑦 ∧ 𝑦 < ( 𝑥 + 1 ) ) ) )
7		riotaex	⊢ ( ℩ 𝑦 ∈ ℤ ( 𝐴 ≤ 𝑦 ∧ 𝑦 < ( 𝐴 + 1 ) ) ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐴 ∈ ℝ → ( ⌈ ‘ 𝐴 ) = ( ℩ 𝑦 ∈ ℤ ( 𝐴 ≤ 𝑦 ∧ 𝑦 < ( 𝐴 + 1 ) ) ) )