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Theorem rlimcn1b 12065
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 2286 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3 rlimcn1b.4 . . . 4  |-  ( ph  ->  F : X --> CC )
43feqmptd 5577 . . 3  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
5 fveq2 5527 . . 3  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5693 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( F `  B ) ) )
7 eqid 2285 . . . 4  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
81, 7fmptd 5686 . . 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 12064 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  ~~> r  ( F `  C ) )
136, 12eqbrtrrd 4047 1  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1686   A.wral 2545   E.wrex 2546   class class class wbr 4025    e. cmpt 4079    o. ccom 4695   -->wf 5253   ` cfv 5257  (class class class)co 5860   CCcc 8737    < clt 8869    - cmin 9039   RR+crp 10356   abscabs 11721    ~~> r crli 11961
This theorem is referenced by:  rlimabs  12084  rlimcj  12085  rlimre  12086  rlimim  12087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-pm 6777  df-rlim 11965
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