Theorem dfres2 5157

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 5132	. 2 |- Rel ( R |` A )
2		relopab 4960	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
3		vex 3048	. . . . 5 |- w e. _V
4	3	brres 5111	. . . 4 |- ( z ( R |` A ) w <-> ( z R w /\ z e. A ) )
5		df-br 4403	. . . 4 |- ( z ( R |` A ) w <-> <. z , w >. e. ( R |` A ) )
6		ancom 452	. . . 4 |- ( ( z R w /\ z e. A ) <-> ( z e. A /\ z R w ) )
7	4, 5, 6	3bitr3i 279	. . 3 |- ( <. z , w >. e. ( R |` A ) <-> ( z e. A /\ z R w ) )
8		vex 3048	. . . 4 |- z e. _V
9		eleq1 2517	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
10		breq1 4405	. . . . 5 |- ( x = z -> ( x R y <-> z R y ) )
11	9, 10	anbi12d 717	. . . 4 |- ( x = z -> ( ( x e. A /\ x R y ) <-> ( z e. A /\ z R y ) ) )
12		breq2 4406	. . . . 5 |- ( y = w -> ( z R y <-> z R w ) )
13	12	anbi2d 710	. . . 4 |- ( y = w -> ( ( z e. A /\ z R y ) <-> ( z e. A /\ z R w ) ) )
14	8, 3, 11, 13	opelopab 4723	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } <-> ( z e. A /\ z R w ) )
15	7, 14	bitr4i 256	. 2 |- ( <. z , w >. e. ( R |` A ) <-> <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } )
16	1, 2, 15	eqrelriiv 4929	1 |- ( R |` A ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   /\ wa 371   = wceq 1444   e. wcel 1887   <.cop 3974   class class class wbr 4402   {copab 4460   |` cres 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-res 4846

This theorem is referenced by:  shftidt2  13144