Theorem ofexg 6249

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.

Step	Hyp	Ref	Expression
1		df-of 6244	. . 3 |- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
2	1	mpofun 6133	. 2 |- Fun oF R
3		resfunexg 5883	. 2 |- ( ( Fun oF R /\ A e. V ) -> ( oF R |` A ) e. _V )
4	2, 3	mpan 424	1 |- ( A e. V -> ( oF R |` A ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2803   i^i cin 3200   |-> cmpt 4155   dom cdm 4731   |` cres 4733   Fun wfun 5327   ` cfv 5333  (class class class)co 6028   oFcof 6242

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-oprab 6032  df-mpo 6033  df-of 6244

This theorem is referenced by:  ofmresex  6308