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Theorem ofexg 6087
Description: A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
Assertion
Ref Expression
ofexg  |-  ( A  e.  V  ->  (  oF 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 6083 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
21mpofun 5977 . 2  |-  Fun  oF R
3 resfunexg 5738 . 2  |-  ( ( Fun  oF R  /\  A  e.  V
)  ->  (  oF R  |`  A )  e.  _V )
42, 3mpan 424 1  |-  ( A  e.  V  ->  (  oF R  |`  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   _Vcvv 2738    i^i cin 3129    |-> cmpt 4065   dom cdm 4627    |` cres 4629   Fun wfun 5211   ` cfv 5217  (class class class)co 5875    oFcof 6081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-oprab 5879  df-mpo 5880  df-of 6083
This theorem is referenced by:  ofmresex  6138
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