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Theorem seinxp 4457
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 4454 . . . . . 6  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
21ancoms 264 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
32rabbidva 2598 . . . 4  |-  ( x  e.  A  ->  { y  e.  A  |  y R x }  =  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x } )
43eleq1d 2151 . . 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 2385 . 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 4116 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
7 df-se 4116 . 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 210 1  |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A
) ) Se  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   A.wral 2353   {crab 2357   _Vcvv 2610    i^i cin 2981   class class class wbr 3805   Se wse 4112    X. cxp 4389
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-se 4116  df-xp 4397
This theorem is referenced by: (None)
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