Theorem funpartss 32051

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

Step	Hyp	Ref	Expression
1		df-funpart 31981	. 2 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2		resss 5422	. 2 |- ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) C_ F
3	1, 2	eqsstri 3635	1 |- Funpart F C_ F

Syntax hints:   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118  Singletoncsingle 31945   Singletonscsingles 31946  Imagecimage 31947  Funpartcfunpart 31956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-res 5126  df-funpart 31981

This theorem is referenced by: (None)