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Theorem ofexg 6124
Description: A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
Assertion
Ref Expression
ofexg  |-  ( A  e.  V  ->  (  o F R  |`  A )  e.  _V )

Proof of Theorem ofexg
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 6120 . . 3  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
21mpt2fun 5988 . 2  |-  Fun  o F R
3 resfunexg 5778 . 2  |-  ( ( Fun  o F R  /\  A  e.  V
)  ->  (  o F R  |`  A )  e.  _V )
42, 3mpan 651 1  |-  ( A  e.  V  ->  (  o F R  |`  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1701   _Vcvv 2822    i^i cin 3185    e. cmpt 4114   dom cdm 4726    |` cres 4728   Fun wfun 5286   ` cfv 5292  (class class class)co 5900    o Fcof 6118
This theorem is referenced by:  ofmresex  6160  psrplusg  16175  dchrplusg  20539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-oprab 5904  df-mpt2 5905  df-of 6120
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