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Theorem seinxp 4936
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 4932 . . . . . 6  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
21ancoms 440 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
32rabbidva 2939 . . . 4  |-  ( x  e.  A  ->  { y  e.  A  |  y R x }  =  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x } )
43eleq1d 2501 . . 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 2729 . 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 4534 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
7 df-se 4534 . 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 269 1  |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A
) ) Se  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    i^i cin 3311   class class class wbr 4204   Se wse 4531    X. cxp 4868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-se 4534  df-xp 4876
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