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Theorem rlimcn1b 12366
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 2431 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3 rlimcn1b.4 . . . 4  |-  ( ph  ->  F : X --> CC )
43feqmptd 5765 . . 3  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
5 fveq2 5714 . . 3  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5887 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( F `  B ) ) )
7 eqid 2430 . . . 4  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
81, 7fmptd 5879 . . 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 12365 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  ~~> r  ( F `  C ) )
136, 12eqbrtrrd 4221 1  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2692   E.wrex 2693   class class class wbr 4199    e. cmpt 4253    o. ccom 4868   -->wf 5436   ` cfv 5440  (class class class)co 6067   CCcc 8972    < clt 9104    - cmin 9275   RR+crp 10596   abscabs 12022    ~~> r crli 12262
This theorem is referenced by:  rlimabs  12385  rlimcj  12386  rlimre  12387  rlimim  12388
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-pm 7007  df-rlim 12266
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