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Theorem rlimmptrcl 13184
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 13078 . . . . 5  |-  ( ( k  e.  A  |->  B )  ~~> r  C  -> 
( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC )
31, 2syl 16 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC )
4 rlimabs.1 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  V )
54ralrimiva 2820 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  V )
6 eqid 2451 . . . . . . 7  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
76fnmpt 5632 . . . . . 6  |-  ( A. k  e.  A  B  e.  V  ->  ( k  e.  A  |->  B )  Fn  A )
8 fndm 5605 . . . . . 6  |-  ( ( k  e.  A  |->  B )  Fn  A  ->  dom  ( k  e.  A  |->  B )  =  A )
95, 7, 83syl 20 . . . . 5  |-  ( ph  ->  dom  ( k  e.  A  |->  B )  =  A )
109feq2d 5642 . . . 4  |-  ( ph  ->  ( ( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC  <->  ( k  e.  A  |->  B ) : A --> CC ) )
113, 10mpbid 210 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
126fmpt 5960 . . 3  |-  ( A. k  e.  A  B  e.  CC  <->  ( k  e.  A  |->  B ) : A --> CC )
1311, 12sylibr 212 . 2  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
1413r19.21bi 2907 1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793   class class class wbr 4387    |-> cmpt 4445   dom cdm 4935    Fn wfn 5508   -->wf 5509   CCcc 9378    ~~> r crli 13062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-pm 7314  df-rlim 13066
This theorem is referenced by:  rlimabs  13185  rlimcj  13186  rlimre  13187  rlimim  13188  rlimadd  13219  rlimsub  13220  rlimmul  13221  rlimdiv  13222  rlimneg  13223  fsumrlim  13373  dvfsumrlim  21616  rlimcxp  22480  cxploglim2  22485
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