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Theorem funpartss 25781
Description: The functional part of  F is a subset of  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartss  |- Funpart F  C_  F

Proof of Theorem funpartss
StepHypRef Expression
1 df-funpart 25710 . 2  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 resss 5162 . 2  |-  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ) 
C_  F
31, 2eqsstri 3370 1  |- Funpart F  C_  F
Colors of variables: wff set class
Syntax hints:   _Vcvv 2948    i^i cin 3311    C_ wss 3312    X. cxp 4868   dom cdm 4870    |` cres 4872    o. ccom 4874  Singletoncsingle 25674   Singletonscsingles 25675  Imagecimage 25676  Funpartcfunpart 25685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-res 4882  df-funpart 25710
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