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Theorem feqresmpt 5857
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 5689 . . . 4  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  ( F  |`  C ) : C --> B )
54feqmptd 5856 . 2  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( ( F  |`  C ) `  x
) ) )
6 fvres 5816 . . 3  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
76mpteq2ia 4485 . 2  |-  ( x  e.  C  |->  ( ( F  |`  C ) `  x ) )  =  ( x  e.  C  |->  ( F `  x
) )
85, 7syl6eq 2511 1  |-  ( ph  ->  ( F  |`  C )  =  ( x  e.  C  |->  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    C_ wss 3439    |-> cmpt 4461    |` cres 4953   -->wf 5525   ` cfv 5529
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537
This theorem is referenced by:  pwfseqlem5  8944  swrd0val  12438  gsumpt  16579  gsumptOLD  16580  dpjidcl  16682  dpjidclOLD  16689  regsumsupp  18180  tsmsxplem2  19863  dvmulbr  21549  dvlip  21601  lhop1lem  21621  loglesqr  22332  jensenlem1  22516  jensen  22518  amgm  22520  gsumle  26411  coinflippv  27030  ftc1cnnclem  28633  dvasin  28648  dvacos  28649  dvreasin  28650  dvreacos  28651  areacirclem1  28652
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