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Theorem rlimcn1b 12063
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimcn1b.1  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  X )
rlimcn1b.2  |-  ( ph  ->  C  e.  X )
rlimcn1b.3  |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C
)
rlimcn1b.4  |-  ( ph  ->  F : X --> CC )
rlimcn1b.5  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  X  ( ( abs `  ( z  -  C
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 C ) ) )  <  x ) )
Assertion
Ref Expression
rlimcn1b  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Distinct variable groups:    x, k,
y, z, A    x, B, y, z    x, C, y, z    k, F, x, y, z    k, X, z    ph, k, x, y
Allowed substitution hints:    ph( z)    B( k)    C( k)    X( x, y)

Proof of Theorem rlimcn1b
StepHypRef Expression
1 rlimcn1b.1 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  X )
2 eqidd 2284 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3 rlimcn1b.4 . . . 4  |-  ( ph  ->  F : X --> CC )
43feqmptd 5575 . . 3  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
5 fveq2 5525 . . 3  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5691 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( F `  B ) ) )
7 eqid 2283 . . . 4  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
81, 7fmptd 5684 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> X )
9 rlimcn1b.2 . . 3  |-  ( ph  ->  C  e.  X )
10 rlimcn1b.3 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  ~~> r  C
)
11 rlimcn1b.5 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  A. z  e.  X  ( ( abs `  ( z  -  C
) )  <  y  ->  ( abs `  (
( F `  z
)  -  ( F `
 C ) ) )  <  x ) )
128, 9, 10, 3, 11rlimcn1 12062 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  ~~> r  ( F `  C ) )
136, 12eqbrtrrd 4045 1  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023    e. cmpt 4077    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719    ~~> r crli 11959
This theorem is referenced by:  rlimabs  12082  rlimcj  12083  rlimre  12084  rlimim  12085
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-csb 3082  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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