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 ) ) )