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 | |- ( ph -> A C_ B ) |
|
o1res2.2 | |- ( ph -> ( x e. B |-> C ) e. O(1) ) |
||
Assertion | o1res2 | |- ( ph -> ( x e. A |-> C ) e. O(1) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimres2.1 | |- ( ph -> A C_ B ) |
|
2 | o1res2.2 | |- ( ph -> ( x e. B |-> C ) e. O(1) ) |
|
3 | 1 | resmptd | |- ( ph -> ( ( x e. B |-> C ) |` A ) = ( x e. A |-> C ) ) |
4 | o1res | |- ( ( x e. B |-> C ) e. O(1) -> ( ( x e. B |-> C ) |` A ) e. O(1) ) |
|
5 | 2 4 | syl | |- ( ph -> ( ( x e. B |-> C ) |` A ) e. O(1) ) |
6 | 3 5 | eqeltrrd | |- ( ph -> ( x e. A |-> C ) e. O(1) ) |