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Theorem rlimmptrcl 12321
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 12215 . . . . 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 2725 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  V )
6 eqid 2380 . . . . . . 7  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
76fnmpt 5504 . . . . . 6  |-  ( A. k  e.  A  B  e.  V  ->  ( k  e.  A  |->  B )  Fn  A )
8 fndm 5477 . . . . . 6  |-  ( ( k  e.  A  |->  B )  Fn  A  ->  dom  ( k  e.  A  |->  B )  =  A )
95, 7, 83syl 19 . . . . 5  |-  ( ph  ->  dom  ( k  e.  A  |->  B )  =  A )
109feq2d 5514 . . . 4  |-  ( ph  ->  ( ( k  e.  A  |->  B ) : dom  ( k  e.  A  |->  B ) --> CC  <->  ( k  e.  A  |->  B ) : A --> CC ) )
113, 10mpbid 202 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
126fmpt 5822 . . 3  |-  ( A. k  e.  A  B  e.  CC  <->  ( k  e.  A  |->  B ) : A --> CC )
1311, 12sylibr 204 . 2  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
1413r19.21bi 2740 1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   class class class wbr 4146    e. cmpt 4200   dom cdm 4811    Fn wfn 5382   -->wf 5383   CCcc 8914    ~~> r crli 12199
This theorem is referenced by:  rlimabs  12322  rlimcj  12323  rlimre  12324  rlimim  12325  rlimadd  12356  rlimsub  12357  rlimmul  12358  rlimdiv  12359  rlimneg  12360  fsumrlim  12510  dvfsumrlim  19775  rlimcxp  20672  cxploglim2  20677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-pm 6950  df-rlim 12203
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