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Theorem rlimcn1b 12270
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 2367 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3 rlimcn1b.4 . . . 4  |-  ( ph  ->  F : X --> CC )
43feqmptd 5682 . . 3  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
5 fveq2 5632 . . 3  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5802 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( F `  B ) ) )
7 eqid 2366 . . . 4  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
81, 7fmptd 5795 . . 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 12269 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  ~~> r  ( F `  C ) )
136, 12eqbrtrrd 4147 1  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715   A.wral 2628   E.wrex 2629   class class class wbr 4125    e. cmpt 4179    o. ccom 4796   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882    < clt 9014    - cmin 9184   RR+crp 10505   abscabs 11926    ~~> r crli 12166
This theorem is referenced by:  rlimabs  12289  rlimcj  12290  rlimre  12291  rlimim  12292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-pm 6918  df-rlim 12170
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