Theorem dfres2 4829

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 4805	. 2 |- Rel ( R |` A )
2		relopab 4626	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
3		vex 2660	. . . . 5 |- w e. _V
4	3	brres 4783	. . . 4 |- ( z ( R |` A ) w <-> ( z R w /\ z e. A ) )
5		df-br 3896	. . . 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 2660	. . . 4 |- z e. _V
9		eleq1 2177	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
10		breq1 3898	. . . . 5 |- ( x = z -> ( x R y <-> z R y ) )
11	9, 10	anbi12d 462	. . . 4 |- ( x = z -> ( ( x e. A /\ x R y ) <-> ( z e. A /\ z R y ) ) )
12		breq2 3899	. . . . 5 |- ( y = w -> ( z R y <-> z R w ) )
13	12	anbi2d 457	. . . 4 |- ( y = w -> ( ( z e. A /\ z R y ) <-> ( z e. A /\ z R w ) ) )
14	8, 3, 11, 13	opelopab 4153	. . 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 4593	1 |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   /\ wa 103   = wceq 1314   e. wcel 1463   <.cop 3496   class class class wbr 3895   {copab 3948   |` cres 4501

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-res 4511

This theorem is referenced by:  shftidt2  10497