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Theorem ofexg 5860
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 5856 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
21mpt2fun 5747 . 2  |-  Fun  oF R
3 resfunexg 5518 . 2  |-  ( ( Fun  oF R  /\  A  e.  V
)  ->  (  oF R  |`  A )  e.  _V )
42, 3mpan 415 1  |-  ( A  e.  V  ->  (  oF R  |`  A )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   _Vcvv 2619    i^i cin 2998    |-> cmpt 3899   dom cdm 4438    |` cres 4440   Fun wfun 5009   ` cfv 5015  (class class class)co 5652    oFcof 5854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-oprab 5656  df-mpt2 5657  df-of 5856
This theorem is referenced by:  ofmresex  5908
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