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Theorem dfres2 4751
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 4728 . 2  |-  Rel  ( R  |`  A )
2 relopab 4552 . 2  |-  Rel  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
3 vex 2622 . . . . 5  |-  w  e. 
_V
43brres 4707 . . . 4  |-  ( z ( R  |`  A ) w  <->  ( z R w  /\  z  e.  A ) )
5 df-br 3838 . . . 4  |-  ( z ( R  |`  A ) w  <->  <. z ,  w >.  e.  ( R  |`  A ) )
6 ancom 262 . . . 4  |-  ( ( z R w  /\  z  e.  A )  <->  ( z  e.  A  /\  z R w ) )
74, 5, 63bitr3i 208 . . 3  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <-> 
( z  e.  A  /\  z R w ) )
8 vex 2622 . . . 4  |-  z  e. 
_V
9 eleq1 2150 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
10 breq1 3840 . . . . 5  |-  ( x  =  z  ->  (
x R y  <->  z R
y ) )
119, 10anbi12d 457 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  x R y )  <-> 
( z  e.  A  /\  z R y ) ) )
12 breq2 3841 . . . . 5  |-  ( y  =  w  ->  (
z R y  <->  z R w ) )
1312anbi2d 452 . . . 4  |-  ( y  =  w  ->  (
( z  e.  A  /\  z R y )  <-> 
( z  e.  A  /\  z R w ) ) )
148, 3, 11, 13opelopab 4089 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) }  <-> 
( z  e.  A  /\  z R w ) )
157, 14bitr4i 185 . 2  |-  ( <.
z ,  w >.  e.  ( R  |`  A )  <->  <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ( x  e.  A  /\  x R y ) } )
161, 2, 15eqrelriiv 4520 1  |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   <.cop 3444   class class class wbr 3837   {copab 3890    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by:  shftidt2  10231
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