Metamath Proof Explorer


Theorem elo1mpt2

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016) (Proof shortened by Mario Carneiro, 26-May-2016)

Ref Expression
Hypotheses elo1mpt.1
|- ( ph -> A C_ RR )
elo1mpt.2
|- ( ( ph /\ x e. A ) -> B e. CC )
elo1d.3
|- ( ph -> C e. RR )
Assertion elo1mpt2
|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )

Proof

Step Hyp Ref Expression
1 elo1mpt.1
 |-  ( ph -> A C_ RR )
2 elo1mpt.2
 |-  ( ( ph /\ x e. A ) -> B e. CC )
3 elo1d.3
 |-  ( ph -> C e. RR )
4 2 lo1o12
 |-  ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
5 2 abscld
 |-  ( ( ph /\ x e. A ) -> ( abs ` B ) e. RR )
6 1 5 3 ello1mpt2
 |-  ( ph -> ( ( x e. A |-> ( abs ` B ) ) e. <_O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )
7 4 6 bitrd
 |-  ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )