Theorem ofexg 6901

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 6897	. . 3 |- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
2	1	mpt2fun 6762	. 2 |- Fun oF R
3		resfunexg 6479	. 2 |- ( ( Fun oF R /\ A e. V ) -> ( oF R |` A ) e. _V )
4	2, 3	mpan 706	1 |- ( A e. V -> ( oF R |` A ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  ofmresex  7165  psrplusg  19381  dchrplusg  24972