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Theorem seinxp 4756
Description: Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
seinxp  |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A
) ) Se  A )

Proof of Theorem seinxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brinxp 4752 . . . . . 6  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
21ancoms 439 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
32rabbidva 2779 . . . 4  |-  ( x  e.  A  ->  { y  e.  A  |  y R x }  =  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x } )
43eleq1d 2349 . . 3  |-  ( x  e.  A  ->  ( { y  e.  A  |  y R x }  e.  _V  <->  { y  e.  A  |  y
( R  i^i  ( A  X.  A ) ) x }  e.  _V ) )
54ralbiia 2575 . 2  |-  ( A. x  e.  A  {
y  e.  A  | 
y R x }  e.  _V  <->  A. x  e.  A  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x }  e.  _V )
6 df-se 4353 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
7 df-se 4353 . 2  |-  ( ( R  i^i  ( A  X.  A ) ) Se  A  <->  A. x  e.  A  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x }  e.  _V )
85, 6, 73bitr4i 268 1  |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A
) ) Se  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151   class class class wbr 4023   Se wse 4350    X. cxp 4687
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-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-se 4353  df-xp 4695
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