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Theorem cncfmptss 29908
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 5256 . . . 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 20587 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . . 5  |-  ( ph  ->  F : A --> B )
76feqmptd 5845 . . . 4  |-  ( ph  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
87reseq1d 5209 . . 3  |-  ( ph  ->  ( F  |`  C )  =  ( ( y  e.  A  |->  ( F `
 y ) )  |`  C ) )
9 nfcv 2613 . . . . . 6  |-  F/_ y F
10 nfcv 2613 . . . . . 6  |-  F/_ y
x
119, 10nffv 5798 . . . . 5  |-  F/_ y
( F `  x
)
12 cncfmptss.1 . . . . . 6  |-  F/_ x F
13 nfcv 2613 . . . . . 6  |-  F/_ x
y
1412, 13nffv 5798 . . . . 5  |-  F/_ x
( F `  y
)
15 fveq2 5791 . . . . 5  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1611, 14, 15cbvmpt 4482 . . . 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 2503 . 2  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( F  |`  C )
)
19 rescncf 20591 . . 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 2539 1  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   F/_wnfc 2599    C_ wss 3428    |-> cmpt 4450    |` cres 4942   -->wf 5514   ` cfv 5518  (class class class)co 6192   -cn->ccncf 20570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-cncf 20572
This theorem is referenced by:  itgsin0pilem1  29930  ibliccsinexp  29931  itgsinexplem1  29934  itgsinexp  29935
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