Description: The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rlimres2.1 | ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) | |
o1res2.2 | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ 𝑂(1) ) | ||
Assertion | o1res2 | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimres2.1 | ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) | |
2 | o1res2.2 | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ 𝑂(1) ) | |
3 | 1 | resmptd | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ↾ 𝐴 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
4 | o1res | ⊢ ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ 𝑂(1) → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ↾ 𝐴 ) ∈ 𝑂(1) ) | |
5 | 2 4 | syl | ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ↾ 𝐴 ) ∈ 𝑂(1) ) |
6 | 3 5 | eqeltrrd | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |