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Theorem feqresmpt 5592
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 5424 . . . 4  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 642 . . 3  |-  ( ph  ->  ( F  |`  C ) : C --> B )
54feqmptd 5591 . 2  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( ( F  |`  C ) `  x
) ) )
6 fvres 5558 . . 3  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
76mpteq2ia 4118 . 2  |-  ( x  e.  C  |->  ( ( F  |`  C ) `  x ) )  =  ( x  e.  C  |->  ( F `  x
) )
85, 7syl6eq 2344 1  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165    e. cmpt 4093    |` cres 4707   -->wf 5267   ` cfv 5271
This theorem is referenced by:  pwfseqlem5  8301  swrd0val  11470  gsumpt  15238  dpjidcl  15309  tsmsxplem2  17852  dvlip  19356  lhop1lem  19376  loglesqr  20114  jensenlem1  20297  jensen  20299  amgm  20301  coinflippv  23699  ftc1cnnclem  25024  dvreasin  25026  dvreacos  25027  areacirclem2  25028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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