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Theorem feqresmpt 5928
Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
feqmptd.1  |-  ( ph  ->  F : A --> B )
feqresmpt.2  |-  ( ph  ->  C  C_  A )
Assertion
Ref Expression
feqresmpt  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Distinct variable groups:    x, A    x, C    x, F
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem feqresmpt
StepHypRef Expression
1 feqmptd.1 . . . 4  |-  ( ph  ->  F : A --> B )
2 feqresmpt.2 . . . 4  |-  ( ph  ->  C  C_  A )
3 fssres 5757 . . . 4  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  ( F  |`  C ) : C --> B )
54feqmptd 5927 . 2  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( ( F  |`  C ) `  x
) ) )
6 fvres 5886 . . 3  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
76mpteq2ia 4535 . 2  |-  ( x  e.  C  |->  ( ( F  |`  C ) `  x ) )  =  ( x  e.  C  |->  ( F `  x
) )
85, 7syl6eq 2524 1  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    C_ wss 3481    |-> cmpt 4511    |` cres 5007   -->wf 5590   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  pwfseqlem5  9053  swrd0val  12628  gsumpt  16861  gsumptOLD  16862  dpjidcl  16979  dpjidclOLD  16986  regsumsupp  18527  tsmsxplem2  20524  dvmulbr  22210  dvlip  22262  lhop1lem  22282  loglesqrt  22998  jensenlem1  23182  jensen  23184  amgm  23186  gsumle  27595  coinflippv  28247  ftc1cnnclem  30015  dvasin  30030  dvacos  30031  dvreasin  30032  dvreacos  30033  areacirclem1  30034  itgperiod  31622  fourierdlem69  31799  fourierdlem73  31803  fourierdlem74  31804  fourierdlem75  31805  fourierdlem76  31806  fourierdlem81  31811  fourierdlem85  31815  fourierdlem88  31818  fourierdlem92  31822  fourierdlem97  31827  fourierdlem100  31830  fourierdlem101  31831  fourierdlem103  31833  fourierdlem104  31834  fourierdlem107  31837  fourierdlem111  31841  fourierdlem112  31842  fouriersw  31855
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