Theorem dfres2 4998

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 4974	. 2 |- Rel ( R |` A )
2		relopab 4792	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
3		vex 2766	. . . . 5 |- w e. _V
4	3	brres 4952	. . . 4 |- ( z ( R |` A ) w <-> ( z R w /\ z e. A ) )
5		df-br 4034	. . . 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 2766	. . . 4 |- z e. _V
9		eleq1 2259	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
10		breq1 4036	. . . . 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 4037	. . . . 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 4306	. . 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 4757	1 |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2167   <.cop 3625   class class class wbr 4033   {copab 4093   |` cres 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-res 4675

This theorem is referenced by:  shftidt2  10997