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Theorem cncfmptid 12789
Description: The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
Assertion
Ref Expression
cncfmptid  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( x  e.  S  |->  x )  e.  ( S -cn-> T ) )
Distinct variable groups:    x, S    x, T

Proof of Theorem cncfmptid
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr 3109 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  S  C_  CC )
2 simpr 109 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  T  C_  CC )
3 simpll 519 . . . . 5  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  x  e.  S
)  ->  S  C_  T
)
4 simpr 109 . . . . 5  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  x  e.  S
)  ->  x  e.  S )
53, 4sseldd 3102 . . . 4  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  x  e.  S
)  ->  x  e.  T )
65fmpttd 5582 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( x  e.  S  |->  x ) : S --> T )
7 simpr 109 . . . 4  |-  ( ( y  e.  S  /\  w  e.  RR+ )  ->  w  e.  RR+ )
87a1i 9 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( ( y  e.  S  /\  w  e.  RR+ )  ->  w  e.  RR+ ) )
9 eqid 2140 . . . . . . . 8  |-  ( x  e.  S  |->  x )  =  ( x  e.  S  |->  x )
10 id 19 . . . . . . . 8  |-  ( x  =  y  ->  x  =  y )
11 simprll 527 . . . . . . . 8  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
y  e.  S )
129, 10, 11, 11fvmptd3 5521 . . . . . . 7  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
( ( x  e.  S  |->  x ) `  y )  =  y )
13 id 19 . . . . . . . 8  |-  ( x  =  z  ->  x  =  z )
14 simprlr 528 . . . . . . . 8  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
z  e.  S )
159, 13, 14, 14fvmptd3 5521 . . . . . . 7  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
( ( x  e.  S  |->  x ) `  z )  =  z )
1612, 15oveq12d 5799 . . . . . 6  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
( ( ( x  e.  S  |->  x ) `
 y )  -  ( ( x  e.  S  |->  x ) `  z ) )  =  ( y  -  z
) )
1716fveq2d 5432 . . . . 5  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
( abs `  (
( ( x  e.  S  |->  x ) `  y )  -  (
( x  e.  S  |->  x ) `  z
) ) )  =  ( abs `  (
y  -  z ) ) )
1817breq1d 3946 . . . 4  |-  ( ( ( S  C_  T  /\  T  C_  CC )  /\  ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ ) )  -> 
( ( abs `  (
( ( x  e.  S  |->  x ) `  y )  -  (
( x  e.  S  |->  x ) `  z
) ) )  < 
w  <->  ( abs `  (
y  -  z ) )  <  w ) )
1918exbiri 380 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( ( ( y  e.  S  /\  z  e.  S )  /\  w  e.  RR+ )  ->  (
( abs `  (
y  -  z ) )  <  w  -> 
( abs `  (
( ( x  e.  S  |->  x ) `  y )  -  (
( x  e.  S  |->  x ) `  z
) ) )  < 
w ) ) )
206, 8, 19elcncf1di 12772 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( ( S  C_  CC  /\  T  C_  CC )  ->  ( x  e.  S  |->  x )  e.  ( S -cn-> T ) ) )
211, 2, 20mp2and 430 1  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( x  e.  S  |->  x )  e.  ( S -cn-> T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1481    C_ wss 3075   class class class wbr 3936    |-> cmpt 3996   ` cfv 5130  (class class class)co 5781   CCcc 7641    < clt 7823    - cmin 7956   RR+crp 9469   abscabs 10800   -cn->ccncf 12763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-map 6551  df-cncf 12764
This theorem is referenced by:  expcncf  12798  dvcnp2cntop  12869
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