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Theorem rlimres 14289
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres |- ( F ~~> r A -> ( F |` B ) ~~> r A )
Proof of Theorem rlimres
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3833 . . . . . . . 8 |- ( dom F i^i B ) C_ dom F
2 ssralv 3666 . . . . . . . 8 |- ( ( dom F i^i B ) C_ dom F -> ( A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) -> A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) )
31, 2ax-mp 5 . . . . . . 7 |- ( A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) -> A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) )
43reximi 3011 . . . . . 6 |- ( E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) -> E. y e. RR A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) )
54ralimi 2952 . . . . 5 |- ( A. x e. RR+ E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) -> A. x e. RR+ E. y e. RR A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) )
65anim2i 593 . . . 4 |- ( ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) -> ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) )
76a1i 11 . . 3 |- ( F ~~> r A -> ( ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) -> ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) ) )
8 rlimf 14232 . . . 4 |- ( F ~~> r A -> F : dom F --> CC )
9 rlimss 14233 . . . 4 |- ( F ~~> r A -> dom F C_ RR )
10 eqidd 2623 . . . 4 |- ( ( F ~~> r A /\ z e. dom F ) -> ( F ` z ) = ( F ` z ) )
118, 9, 10rlim 14226 . . 3 |- ( F ~~> r A -> ( F ~~> r A <-> ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. dom F ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) ) )
12 fssres 6070 . . . . . 6 |- ( ( F : dom F --> CC /\ ( dom F i^i B ) C_ dom F ) -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> CC )
138, 1, 12sylancl 694 . . . . 5 |- ( F ~~> r A -> ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> CC )
14 resres 5409 . . . . . . 7 |- ( ( F |` dom F ) |` B ) = ( F |` ( dom F i^i B ) )
15 ffn 6045 . . . . . . . . 9 |- ( F : dom F --> CC -> F Fn dom F )
16 fnresdm 6000 . . . . . . . . 9 |- ( F Fn dom F -> ( F |` dom F ) = F )
178, 15, 163syl 18 . . . . . . . 8 |- ( F ~~> r A -> ( F |` dom F ) = F )
1817reseq1d 5395 . . . . . . 7 |- ( F ~~> r A -> ( ( F |` dom F ) |` B ) = ( F |` B ) )
1914, 18syl5eqr 2670 . . . . . 6 |- ( F ~~> r A -> ( F |` ( dom F i^i B ) ) = ( F |` B ) )
2019feq1d 6030 . . . . 5 |- ( F ~~> r A -> ( ( F |` ( dom F i^i B ) ) : ( dom F i^i B ) --> CC <-> ( F |` B ) : ( dom F i^i B ) --> CC ) )
2113, 20mpbid 222 . . . 4 |- ( F ~~> r A -> ( F |` B ) : ( dom F i^i B ) --> CC )
221, 9syl5ss 3614 . . . 4 |- ( F ~~> r A -> ( dom F i^i B ) C_ RR )
23 inss2 3834 . . . . . . 7 |- ( dom F i^i B ) C_ B
2423sseli 3599 . . . . . 6 |- ( z e. ( dom F i^i B ) -> z e. B )
25 fvres 6207 . . . . . 6 |- ( z e. B -> ( ( F |` B ) ` z ) = ( F ` z ) )
2624, 25syl 17 . . . . 5 |- ( z e. ( dom F i^i B ) -> ( ( F |` B ) ` z ) = ( F ` z ) )
2726adantl 482 . . . 4 |- ( ( F ~~> r A /\ z e. ( dom F i^i B ) ) -> ( ( F |` B ) ` z ) = ( F ` z ) )
2821, 22, 27rlim 14226 . . 3 |- ( F ~~> r A -> ( ( F |` B ) ~~> r A <-> ( A e. CC /\ A. x e. RR+ E. y e. RR A. z e. ( dom F i^i B ) ( y <_ z -> ( abs ` ( ( F ` z ) - A ) ) < x ) ) ) )
297, 11, 283imtr4d 283 . 2 |- ( F ~~> r A -> ( F ~~> r A -> ( F |` B ) ~~> r A ) )
3029pm2.43i 52 1 |- ( F ~~> r A -> ( F |` B ) ~~> r A )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   abscabs 13974   ~~> r crli 14216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-rlim 14220
This theorem is referenced by:  rlimres2  14292  pnt  25303
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