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Theorem cncfmptss 29613
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 5144 . . . 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 20311 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . . 5  |-  ( ph  ->  F : A --> B )
76feqmptd 5732 . . . 4  |-  ( ph  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
87reseq1d 5096 . . 3  |-  ( ph  ->  ( F  |`  C )  =  ( ( y  e.  A  |->  ( F `
 y ) )  |`  C ) )
9 nfcv 2569 . . . . . 6  |-  F/_ y F
10 nfcv 2569 . . . . . 6  |-  F/_ y
x
119, 10nffv 5686 . . . . 5  |-  F/_ y
( F `  x
)
12 cncfmptss.1 . . . . . 6  |-  F/_ x F
13 nfcv 2569 . . . . . 6  |-  F/_ x
y
1412, 13nffv 5686 . . . . 5  |-  F/_ x
( F `  y
)
15 fveq2 5679 . . . . 5  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1611, 14, 15cbvmpt 4370 . . . 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 2476 . 2  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( F  |`  C )
)
19 rescncf 20315 . . 3  |-  ( C 
C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
201, 4, 19sylc 60 . 2  |-  ( ph  ->  ( F  |`  C )  e.  ( C -cn-> B ) )
2118, 20eqeltrd 2507 1  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    e. wcel 1755   F/_wnfc 2556    C_ wss 3316    e. cmpt 4338    |` cres 4829   -->wf 5402   ` cfv 5406  (class class class)co 6080   -cn->ccncf 20294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-cncf 20296
This theorem is referenced by:  itgsin0pilem1  29636  ibliccsinexp  29637  itgsinexplem1  29640  itgsinexp  29641
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