Theorem lo1mptrcl

Description: Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	o1add2.1	|- ( ( ph /\ x e. A ) -> B e. V )
	lo1mptrcl.3	|- ( ph -> ( x e. A |-> B ) e. <_O(1) )
Assertion	lo1mptrcl	|- ( ( ph /\ x e. A ) -> B e. RR )

Proof

Step	Hyp	Ref	Expression
1		o1add2.1	|- ( ( ph /\ x e. A ) -> B e. V )
2		lo1mptrcl.3	|- ( ph -> ( x e. A |-> B ) e. <_O(1) )
3		lo1f	|- ( ( x e. A |-> B ) e. <_O(1) -> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> RR )
4	2 3	syl	|- ( ph -> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> RR )
5	1	ralrimiva	|- ( ph -> A. x e. A B e. V )
6		dmmptg	|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
7	5 6	syl	|- ( ph -> dom ( x e. A |-> B ) = A )
8	7	feq2d	|- ( ph -> ( ( x e. A |-> B ) : dom ( x e. A |-> B ) --> RR <-> ( x e. A |-> B ) : A --> RR ) )
9	4 8	mpbid	|- ( ph -> ( x e. A |-> B ) : A --> RR )
10	9	fvmptelrn	|- ( ( ph /\ x e. A ) -> B e. RR )