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Theorem rlimmptrcl 12081
Description: Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimabs.1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  V )
rlimabs.2  |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C
)
Assertion
Ref Expression
rlimmptrcl  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Distinct variable groups:    A, k    ph, k
Allowed substitution hints:    B( k)    C( k)    V( k)

Proof of Theorem rlimmptrcl
StepHypRef Expression
1 rlimabs.2 . . . . 5  |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C
)
2 rlimf 11975 . . . . 5  |-  ( ( k  e.  A  |->  B )  ~~> r  C  -> 
( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC )
31, 2syl 15 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC )
4 rlimabs.1 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  V )
54ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  V )
6 eqid 2283 . . . . . . 7  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
76fnmpt 5370 . . . . . 6  |-  ( A. k  e.  A  B  e.  V  ->  ( k  e.  A  |->  B )  Fn  A )
8 fndm 5343 . . . . . 6  |-  ( ( k  e.  A  |->  B )  Fn  A  ->  dom  ( k  e.  A  |->  B )  =  A )
95, 7, 83syl 18 . . . . 5  |-  ( ph  ->  dom  ( k  e.  A  |->  B )  =  A )
109feq2d 5380 . . . 4  |-  ( ph  ->  ( ( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC  <->  ( k  e.  A  |->  B ) : A --> CC ) )
113, 10mpbid 201 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
126fmpt 5681 . . 3  |-  ( A. k  e.  A  B  e.  CC  <->  ( k  e.  A  |->  B ) : A --> CC )
1311, 12sylibr 203 . 2  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
1413r19.21bi 2641 1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    e. cmpt 4077   dom cdm 4689    Fn wfn 5250   -->wf 5251   CCcc 8735    ~~> r crli 11959
This theorem is referenced by:  rlimabs  12082  rlimcj  12083  rlimre  12084  rlimim  12085  rlimadd  12116  rlimsub  12117  rlimmul  12118  rlimdiv  12119  rlimneg  12120  fsumrlim  12269  dvfsumrlim  19378  rlimcxp  20268  cxploglim2  20273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pm 6775  df-rlim 11963
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