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 biimpi 
 |-  ( F e. dom ~~> -> F ~~> ( ~~> ` F ) )
8 7 adantl 
 |-  ( ( ph /\ F e. dom ~~> ) -> F ~~> ( ~~> ` F ) )
9 1 adantr 
 |-  ( ( ph /\ F e. dom ~~> ) -> M e. ZZ )
10 simpr 
 |-  ( ( ph /\ F e. dom ~~> ) -> F e. dom ~~> )
11 9 10 climresd 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` F ) <-> F ~~> ( ~~> ` F ) ) )
12 8 11 mpbird 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` F ) )
13 4 5 12 breldmd 
 |-  ( ( ph /\ F e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) e. dom ~~> )
14 2 adantr 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F e. V )
15 fvexd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) e. _V )
16 climdm 
 |-  ( ( F |` ( ZZ>= ` M ) ) e. dom ~~> <-> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
17 16 biimpi 
 |-  ( ( F |` ( ZZ>= ` M ) ) e. dom ~~> -> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
18 17 adantl 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
19 1 adantr 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> M e. ZZ )
20 19 14 climresd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> ( ( F |` ( ZZ>= ` M ) ) ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) <-> F ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) ) )
21 18 20 mpbid 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F ~~> ( ~~> ` ( F |` ( ZZ>= ` M ) ) ) )
22 14 15 21 breldmd 
 |-  ( ( ph /\ ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) -> F e. dom ~~> )
23 13 22 impbida 
 |-  ( ph -> ( F e. dom ~~> <-> ( F |` ( ZZ>= ` M ) ) e. dom ~~> ) )