MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seinxp Unicode version

Theorem seinxp 4884
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 4880 . . . . . 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 2890 . . . 4  |-  ( x  e.  A  ->  { y  e.  A  |  y R x }  =  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x } )
43eleq1d 2453 . . 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 2681 . 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 4483 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
7 df-se 4483 . 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 1717   A.wral 2649   {crab 2653   _Vcvv 2899    i^i cin 3262   class class class wbr 4153   Se wse 4480    X. cxp 4816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-se 4483  df-xp 4824
  Copyright terms: Public domain W3C validator