Theorem dfres2 4930

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 4906	. 2 |- Rel ( R |` A )
2		relopab 4725	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
3		vex 2724	. . . . 5 |- w e. _V
4	3	brres 4884	. . . 4 |- ( z ( R |` A ) w <-> ( z R w /\ z e. A ) )
5		df-br 3977	. . . 4 |- ( z ( R |` A ) w <-> <. z , w >. e. ( R |` A ) )
6		ancom 264	. . . 4 |- ( ( z R w /\ z e. A ) <-> ( z e. A /\ z R w ) )
7	4, 5, 6	3bitr3i 209	. . 3 |- ( <. z , w >. e. ( R |` A ) <-> ( z e. A /\ z R w ) )
8		vex 2724	. . . 4 |- z e. _V
9		eleq1 2227	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
10		breq1 3979	. . . . 5 |- ( x = z -> ( x R y <-> z R y ) )
11	9, 10	anbi12d 465	. . . 4 |- ( x = z -> ( ( x e. A /\ x R y ) <-> ( z e. A /\ z R y ) ) )
12		breq2 3980	. . . . 5 |- ( y = w -> ( z R y <-> z R w ) )
13	12	anbi2d 460	. . . 4 |- ( y = w -> ( ( z e. A /\ z R y ) <-> ( z e. A /\ z R w ) ) )
14	8, 3, 11, 13	opelopab 4243	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } <-> ( z e. A /\ z R w ) )
15	7, 14	bitr4i 186	. 2 |- ( <. z , w >. e. ( R |` A ) <-> <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } )
16	1, 2, 15	eqrelriiv 4692	1 |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   /\ wa 103   = wceq 1342   e. wcel 2135   <.cop 3573   class class class wbr 3976   {copab 4036   |` cres 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-rel 4605  df-res 4610

This theorem is referenced by:  shftidt2  10760