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Theorem ofexg 6437
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 6433 . . 3  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
21mpt2fun 6305 . 2  |-  Fun  oF R
3 resfunexg 6055 . 2  |-  ( ( Fun  oF R  /\  A  e.  V
)  ->  (  oF R  |`  A )  e.  _V )
42, 3mpan 670 1  |-  ( A  e.  V  ->  (  oF R  |`  A )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3078    i^i cin 3438    |-> cmpt 4461   dom cdm 4951    |` cres 4953   Fun wfun 5523   ` cfv 5529  (class class class)co 6203    oFcof 6431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-oprab 6207  df-mpt2 6208  df-of 6433
This theorem is referenced by:  ofmresex  6687  psrplusg  17585  dchrplusg  22729
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