Metamath Proof Explorer

Theorem climresdm

Description: A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023)

Ref Expression
Hypotheses climresdm.1 
|- ( ph -> M e. ZZ )
climresdm.2 
|- ( ph -> F e. V )
Assertion climresdm 
|- ( ph -> ( F e. dom ~~> <-> ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) )

Proof

Step Hyp Ref Expression
1 climresdm.1 
 |-  ( ph -> M e. ZZ )
2 climresdm.2 
 |-  ( ph -> F e. V )
3 resexg 
 |-  ( F e. dom ~~> -> ( F |` ( ZZ>= ` M ) ) e. _V )
4 3 adantl 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) e. _V )
5 fvexd 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( ~~> ` F ) e. _V )
6 climdm 
 |-  ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
7 6 bilani 
 |-  ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` F ) )
8 1 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> M e. ZZ )
9 simpr 
 |-  ( ( ph /\ F e. dom ~~> ) -> F e. dom ~~> )
10 8 9 climresd 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` F ) <-> F ~~> ( ~~> ` F ) ) )
11 7 10 mpbird 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` F ) )
12 4 5 11 breldmd 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) e. dom ~~> )
13 2 adantr 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F e. V )
14 fvexd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) e. _V )
15 climdm 
 |-  ( ( F |` ( ZZ>= ` M ) ) e. dom ~~> <-> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
16 15 bilani 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
17 1 adantr 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> M e. ZZ )
18 17 13 climresd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) <-> F ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) ) )
19 16 18 mpbid 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
20 13 14 19 breldmd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F e. dom ~~> )
21 12 20 impbida 
 |-  ( ph -> ( F e. dom ~~> <-> ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) )