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Theorem cncfmptss 27592
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 5158 . . . 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 18884 . . . . . 6  |-  ( F  e.  ( A -cn-> B )  ->  F : A
--> B )
64, 5syl 16 . . . . 5  |-  ( ph  ->  F : A --> B )
76feqmptd 5746 . . . 4  |-  ( ph  ->  F  =  ( y  e.  A  |->  ( F `
 y ) ) )
87reseq1d 5112 . . 3  |-  ( ph  ->  ( F  |`  C )  =  ( ( y  e.  A  |->  ( F `
 y ) )  |`  C ) )
9 nfcv 2548 . . . . . 6  |-  F/_ y F
10 nfcv 2548 . . . . . 6  |-  F/_ y
x
119, 10nffv 5702 . . . . 5  |-  F/_ y
( F `  x
)
12 cncfmptss.1 . . . . . 6  |-  F/_ x F
13 nfcv 2548 . . . . . 6  |-  F/_ x
y
1412, 13nffv 5702 . . . . 5  |-  F/_ x
( F `  y
)
15 fveq2 5695 . . . . 5  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1611, 14, 15cbvmpt 4267 . . . 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 2455 . 2  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  =  ( F  |`  C )
)
19 rescncf 18888 . . 3  |-  ( C 
C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
201, 4, 19sylc 58 . 2  |-  ( ph  ->  ( F  |`  C )  e.  ( C -cn-> B ) )
2118, 20eqeltrd 2486 1  |-  ( ph  ->  ( x  e.  C  |->  ( F `  x
) )  e.  ( C -cn-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   F/_wnfc 2535    C_ wss 3288    e. cmpt 4234    |` cres 4847   -->wf 5417   ` cfv 5421  (class class class)co 6048   -cn->ccncf 18867
This theorem is referenced by:  itgsin0pilem1  27619  ibliccsinexp  27620  itgsinexplem1  27623  itgsinexp  27624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-cncf 18869
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