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Theorem cncfmptss 27731
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 . . . 4  |-  ( ph  ->  C  C_  A )
2 resmpt 5220 . . . 4  |-  ( C 
C_  A  ->  (
( y  e.  A  |->  ( F `  y
) )  |`  C )  =  ( y  e.  C  |->  ( F `  y ) ) )
31, 2syl 16 . . 3  |-  ( ph  ->  ( ( y  e.  A  |->  ( F `  y ) )  |`  C )  =  ( y  e.  C  |->  ( F `  y ) ) )
4 cncfmptss.2 . . . . . 6  |-  ( ph  ->  F  e.  ( A
-cn-> B ) )
5 cncff 18954 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . . 5  |-  ( ph  ->  F : A --> B )
76feqmptd 5808 . . . 4  |-  ( ph  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
87reseq1d 5174 . . 3  |-  ( ph  ->  ( F  |`  C )  =  ( ( y  e.  A  |->  ( F `
 y ) )  |`  C ) )
9 nfcv 2578 . . . . . 6  |-  F/_ y F
10 nfcv 2578 . . . . . 6  |-  F/_ y
x
119, 10nffv 5764 . . . . 5  |-  F/_ y
( F `  x
)
12 cncfmptss.1 . . . . . 6  |-  F/_ x F
13 nfcv 2578 . . . . . 6  |-  F/_ x
y
1412, 13nffv 5764 . . . . 5  |-  F/_ x
( F `  y
)
15 fveq2 5757 . . . . 5  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1611, 14, 15cbvmpt 4324 . . . 4  |-  ( x  e.  C  |->  ( F `
 x ) )  =  ( y  e.  C  |->  ( F `  y ) )
1716a1i 11 . . 3  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( y  e.  C  |->  ( F `  y ) ) )
183, 8, 173eqtr4rd 2485 . 2  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( F  |`  C )
)
19 rescncf 18958 . . 3  |-  ( C 
C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
201, 4, 19sylc 59 . 2  |-  ( ph  ->  ( F  |`  C )  e.  ( C -cn-> B ) )
2118, 20eqeltrd 2516 1  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   F/_wnfc 2565    C_ wss 3306    e. cmpt 4291    |` cres 4909   -->wf 5479   ` cfv 5483  (class class class)co 6110   -cn->ccncf 18937
This theorem is referenced by:  itgsin0pilem1  27758  ibliccsinexp  27759  itgsinexplem1  27762  itgsinexp  27763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-map 7049  df-cncf 18939
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