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Theorem dfres2 5018
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.
StepHypRef Expression
1 relres 4999 . 2  |-  Rel  ( R  |`  A )
2 relopab 4828 . 2  |-  Rel  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
3 vex 2804 . . . . 5  |-  w  e. 
_V
43brres 4977 . . . 4  |-  ( z ( R  |`  A ) w  <->  ( z R w  /\  z  e.  A ) )
5 df-br 4040 . . . 4  |-  ( z ( R  |`  A ) w  <->  <. z ,  w >.  e.  ( R  |`  A ) )
6 ancom 437 . . . 4  |-  ( ( z R w  /\  z  e.  A )  <->  ( z  e.  A  /\  z R w ) )
74, 5, 63bitr3i 266 . . 3  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <-> 
( z  e.  A  /\  z R w ) )
8 vex 2804 . . . 4  |-  z  e. 
_V
9 eleq1 2356 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
10 breq1 4042 . . . . 5  |-  ( x  =  z  ->  (
x R y  <->  z R
y ) )
119, 10anbi12d 691 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  x R y )  <-> 
( z  e.  A  /\  z R y ) ) )
12 breq2 4043 . . . . 5  |-  ( y  =  w  ->  (
z R y  <->  z R w ) )
1312anbi2d 684 . . . 4  |-  ( y  =  w  ->  (
( z  e.  A  /\  z R y )  <-> 
( z  e.  A  /\  z R w ) ) )
148, 3, 11, 13opelopab 4302 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) }  <-> 
( z  e.  A  /\  z R w ) )
157, 14bitr4i 243 . 2  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <->  <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) } )
161, 2, 15eqrelriiv 4797 1  |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   {copab 4092    |` cres 4707
This theorem is referenced by:  shftidt2  11592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-res 4717
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