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Theorem dfres2 5133
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 5114 . 2  |-  Rel  ( R  |`  A )
2 relopab 4941 . 2  |-  Rel  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
3 vex 2902 . . . . 5  |-  w  e. 
_V
43brres 5092 . . . 4  |-  ( z ( R  |`  A ) w  <->  ( z R w  /\  z  e.  A ) )
5 df-br 4154 . . . 4  |-  ( z ( R  |`  A ) w  <->  <. z ,  w >.  e.  ( R  |`  A ) )
6 ancom 438 . . . 4  |-  ( ( z R w  /\  z  e.  A )  <->  ( z  e.  A  /\  z R w ) )
74, 5, 63bitr3i 267 . . 3  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <-> 
( z  e.  A  /\  z R w ) )
8 vex 2902 . . . 4  |-  z  e. 
_V
9 eleq1 2447 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
10 breq1 4156 . . . . 5  |-  ( x  =  z  ->  (
x R y  <->  z R
y ) )
119, 10anbi12d 692 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  x R y )  <-> 
( z  e.  A  /\  z R y ) ) )
12 breq2 4157 . . . . 5  |-  ( y  =  w  ->  (
z R y  <->  z R w ) )
1312anbi2d 685 . . . 4  |-  ( y  =  w  ->  (
( z  e.  A  /\  z R y )  <-> 
( z  e.  A  /\  z R w ) ) )
148, 3, 11, 13opelopab 4417 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) }  <-> 
( z  e.  A  /\  z R w ) )
157, 14bitr4i 244 . 2  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <->  <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) } )
161, 2, 15eqrelriiv 4910 1  |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3760   class class class wbr 4153   {copab 4206    |` cres 4820
This theorem is referenced by:  shftidt2  11823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-res 4830
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