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Theorem rlimcn1b 12079
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 2297 . . 3  |-  ( ph  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B ) )
3 rlimcn1b.4 . . . 4  |-  ( ph  ->  F : X --> CC )
43feqmptd 5591 . . 3  |-  ( ph  ->  F  =  ( z  e.  X  |->  ( F `
 z ) ) )
5 fveq2 5541 . . 3  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
61, 2, 4, 5fmptco 5707 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  =  ( k  e.  A  |->  ( F `  B ) ) )
7 eqid 2296 . . . 4  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
81, 7fmptd 5700 . . 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 12078 . 2  |-  ( ph  ->  ( F  o.  (
k  e.  A  |->  B ) )  ~~> r  ( F `  C ) )
136, 12eqbrtrrd 4061 1  |-  ( ph  ->  ( k  e.  A  |->  ( F `  B
) )  ~~> r  ( F `  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    < clt 8883    - cmin 9053   RR+crp 10370   abscabs 11735    ~~> r crli 11975
This theorem is referenced by:  rlimabs  12098  rlimcj  12099  rlimre  12100  rlimim  12101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-rlim 11979
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