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Metamath Proof Explorer

Theorem ello1

Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016)

Ref Expression
Assertion ello1 ( 𝐹 ∈ ≤𝑂(1) ↔ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )

Proof

Step Hyp Ref Expression
1 dmeq ⊢ ( 𝑓 = 𝐹 → dom 𝑓 = dom 𝐹 )
2 1 ineq1d ⊢ ( 𝑓 = 𝐹 → ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) = ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) )
3 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
4 3 breq1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 ↔ ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )
5 2 4 raleqbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 ↔ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )
6 5 2rexbidv ⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 ↔ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )
7 df-lo1 ⊢ ≤𝑂(1) = { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓 ‘ 𝑦 ) ≤ 𝑚 }
8 6 7 elrab2 ⊢ ( 𝐹 ∈ ≤𝑂(1) ↔ ( 𝐹 ∈ ( ℝ ↑pm ℝ ) ∧ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝐹 ∩ ( 𝑥 [,) +∞ ) ) ( 𝐹 ‘ 𝑦 ) ≤ 𝑚 ) )