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Theorem rlimmptrcl 12391
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 12285 . . . . 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 2781 . . . . . 6  |-  ( ph  ->  A. k  e.  A  B  e.  V )
6 eqid 2435 . . . . . . 7  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
76fnmpt 5563 . . . . . 6  |-  ( A. k  e.  A  B  e.  V  ->  ( k  e.  A  |->  B )  Fn  A )
8 fndm 5536 . . . . . 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 5573 . . . 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 5882 . . 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 2796 1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    e. cmpt 4258   dom cdm 4870    Fn wfn 5441   -->wf 5442   CCcc 8978    ~~> r crli 12269
This theorem is referenced by:  rlimabs  12392  rlimcj  12393  rlimre  12394  rlimim  12395  rlimadd  12426  rlimsub  12427  rlimmul  12428  rlimdiv  12429  rlimneg  12430  fsumrlim  12580  dvfsumrlim  19905  rlimcxp  20802  cxploglim2  20807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pm 7013  df-rlim 12273
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