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Metamath Proof Explorer

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 fvmptelcdm 
 |-  ( ( ph /\ x e. A ) -> B e. RR )