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Theorem dfres2 5002
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 4983 . 2  |-  Rel  ( R  |`  A )
2 relopab 4812 . 2  |-  Rel  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
3 vex 2791 . . . . 5  |-  w  e. 
_V
43brres 4961 . . . 4  |-  ( z ( R  |`  A ) w  <->  ( z R w  /\  z  e.  A ) )
5 df-br 4024 . . . 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 2791 . . . 4  |-  z  e. 
_V
9 eleq1 2343 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
10 breq1 4026 . . . . 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 4027 . . . . 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 4286 . . 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 4781 1  |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   {copab 4076    |` cres 4691
This theorem is referenced by:  shftidt2  11576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
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