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Theorem dfres2 5163
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 5138 . 2  |-  Rel  ( R  |`  A )
2 relopab 4965 . 2  |-  Rel  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
3 vex 3034 . . . . 5  |-  w  e. 
_V
43brres 5117 . . . 4  |-  ( z ( R  |`  A ) w  <->  ( z R w  /\  z  e.  A ) )
5 df-br 4396 . . . 4  |-  ( z ( R  |`  A ) w  <->  <. z ,  w >.  e.  ( R  |`  A ) )
6 ancom 457 . . . 4  |-  ( ( z R w  /\  z  e.  A )  <->  ( z  e.  A  /\  z R w ) )
74, 5, 63bitr3i 283 . . 3  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <-> 
( z  e.  A  /\  z R w ) )
8 vex 3034 . . . 4  |-  z  e. 
_V
9 eleq1 2537 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
10 breq1 4398 . . . . 5  |-  ( x  =  z  ->  (
x R y  <->  z R
y ) )
119, 10anbi12d 725 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  x R y )  <-> 
( z  e.  A  /\  z R y ) ) )
12 breq2 4399 . . . . 5  |-  ( y  =  w  ->  (
z R y  <->  z R w ) )
1312anbi2d 718 . . . 4  |-  ( y  =  w  ->  (
( z  e.  A  /\  z R y )  <-> 
( z  e.  A  /\  z R w ) ) )
148, 3, 11, 13opelopab 4723 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) }  <-> 
( z  e.  A  /\  z R w ) )
157, 14bitr4i 260 . 2  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <->  <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) } )
161, 2, 15eqrelriiv 4934 1  |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452    e. wcel 1904   <.cop 3965   class class class wbr 4395   {copab 4453    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-res 4851
This theorem is referenced by:  shftidt2  13221
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