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Theorem feqresmpt 5902
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 )
31, 2fssresd 5734 . . 3  |-  ( ph  ->  ( F  |`  C ) : C --> B )
43feqmptd 5901 . 2  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( ( F  |`  C ) `  x
) ) )
5 fvres 5862 . . 3  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
65mpteq2ia 4521 . 2  |-  ( x  e.  C  |->  ( ( F  |`  C ) `  x ) )  =  ( x  e.  C  |->  ( F `  x
) )
74, 6syl6eq 2511 1  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    C_ wss 3461    |-> cmpt 4497    |` cres 4990   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  pwfseqlem5  9030  swrd0val  12640  gsumpt  17187  gsumptOLD  17188  dpjidcl  17305  dpjidclOLD  17312  regsumsupp  18834  tsmsxplem2  20825  dvmulbr  22511  dvlip  22563  lhop1lem  22583  loglesqrt  23303  jensenlem1  23517  jensen  23519  amgm  23521  gsumle  28007  coinflippv  28689  ftc1cnnclem  30331  dvasin  30346  dvacos  30347  dvreasin  30348  dvreacos  30349  areacirclem1  30350  itgperiod  32022  fourierdlem69  32200  fourierdlem73  32204  fourierdlem74  32205  fourierdlem75  32206  fourierdlem76  32207  fourierdlem81  32212  fourierdlem85  32216  fourierdlem88  32219  fourierdlem92  32223  fourierdlem97  32228  fourierdlem100  32231  fourierdlem101  32232  fourierdlem103  32234  fourierdlem104  32235  fourierdlem107  32238  fourierdlem111  32242  fourierdlem112  32243  fouriersw  32256  pfxres  32635
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