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Theorem cncfmptss 27717
Description: A continuous complex function restricted to a subset is continuous, using "map to" notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
cncfmptss.1  |-  F/_ x F
cncfmptss.2  |-  ( ph  ->  F  e.  ( A
-cn-> B ) )
cncfmptss.3  |-  ( ph  ->  C  C_  A )
Assertion
Ref Expression
cncfmptss  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Distinct variable group:    x, C
Allowed substitution hints:    ph( x)    A( x)    B( x)    F( x)

Proof of Theorem cncfmptss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cncfmptss.3 . . . . 5  |-  ( ph  ->  C  C_  A )
2 resmpt 5000 . . . . 5  |-  ( C 
C_  A  ->  (
( y  e.  A  |->  ( F `  y
) )  |`  C )  =  ( y  e.  C  |->  ( F `  y ) ) )
31, 2syl 15 . . . 4  |-  ( ph  ->  ( ( y  e.  A  |->  ( F `  y ) )  |`  C )  =  ( y  e.  C  |->  ( F `  y ) ) )
4 cncfmptss.2 . . . . . . 7  |-  ( ph  ->  F  e.  ( A
-cn-> B ) )
5 cncff 18397 . . . . . . 7  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  F : A --> B )
76feqmptd 5575 . . . . 5  |-  ( ph  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
87reseq1d 4954 . . . 4  |-  ( ph  ->  ( F  |`  C )  =  ( ( y  e.  A  |->  ( F `
 y ) )  |`  C ) )
9 nfcv 2419 . . . . . . 7  |-  F/_ y F
10 nfcv 2419 . . . . . . 7  |-  F/_ y
x
119, 10nffv 5532 . . . . . 6  |-  F/_ y
( F `  x
)
12 cncfmptss.1 . . . . . . 7  |-  F/_ x F
13 nfcv 2419 . . . . . . 7  |-  F/_ x
y
1412, 13nffv 5532 . . . . . 6  |-  F/_ x
( F `  y
)
15 fveq2 5525 . . . . . 6  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1611, 14, 15cbvmpt 4110 . . . . 5  |-  ( x  e.  C  |->  ( F `
 x ) )  =  ( y  e.  C  |->  ( F `  y ) )
1716a1i 10 . . . 4  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( y  e.  C  |->  ( F `  y ) ) )
183, 8, 173eqtr4d 2325 . . 3  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
1918eqcomd 2288 . 2  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( F  |`  C )
)
20 rescncf 18401 . . 3  |-  ( C 
C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
211, 4, 20sylc 56 . 2  |-  ( ph  ->  ( F  |`  C )  e.  ( C -cn-> B ) )
2219, 21eqeltrd 2357 1  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   F/_wnfc 2406    C_ wss 3152    e. cmpt 4077    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   -cn->ccncf 18380
This theorem is referenced by:  itgsin0pilem1  27744  ibliccsinexp  27745  itgsinexplem1  27748  itgsinexp  27749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cncf 18382
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