Theorem dfres2 4957

Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Assertion

Ref	Expression
dfres2	|- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Distinct variable groups:   x, y, A   x, R, y

Proof of Theorem dfres2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4933	. 2 |- Rel ( R |` A )
2		relopab 4752	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
3		vex 2740	. . . . 5 |- w e. _V
4	3	brres 4911	. . . 4 |- ( z ( R |` A ) w <-> ( z R w /\ z e. A ) )
5		df-br 4003	. . . 4 |- ( z ( R |` A ) w <-> <. z , w >. e. ( R |` A ) )
6		ancom 266	. . . 4 |- ( ( z R w /\ z e. A ) <-> ( z e. A /\ z R w ) )
7	4, 5, 6	3bitr3i 210	. . 3 |- ( <. z , w >. e. ( R |` A ) <-> ( z e. A /\ z R w ) )
8		vex 2740	. . . 4 |- z e. _V
9		eleq1 2240	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
10		breq1 4005	. . . . 5 |- ( x = z -> ( x R y <-> z R y ) )
11	9, 10	anbi12d 473	. . . 4 |- ( x = z -> ( ( x e. A /\ x R y ) <-> ( z e. A /\ z R y ) ) )
12		breq2 4006	. . . . 5 |- ( y = w -> ( z R y <-> z R w ) )
13	12	anbi2d 464	. . . 4 |- ( y = w -> ( ( z e. A /\ z R y ) <-> ( z e. A /\ z R w ) ) )
14	8, 3, 11, 13	opelopab 4270	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } <-> ( z e. A /\ z R w ) )
15	7, 14	bitr4i 187	. 2 |- ( <. z , w >. e. ( R |` A ) <-> <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } )
16	1, 2, 15	eqrelriiv 4719	1 |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   <.cop 3595   class class class wbr 4002   {copab 4062   |` cres 4627

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-sep 4120  ax-pow 4173  ax-pr 4208

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-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-res 4637

This theorem is referenced by:  shftidt2  10833