ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seinxp Unicode version

Theorem seinxp 4580
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 4577 . . . . . 6  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
21ancoms 266 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
32rabbidva 2648 . . . 4  |-  ( x  e.  A  ->  { y  e.  A  |  y R x }  =  { y  e.  A  |  y ( R  i^i  ( A  X.  A ) ) x } )
43eleq1d 2186 . . 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 2426 . 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 4225 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
7 df-se 4225 . 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 211 1  |-  ( R Se  A  <->  ( R  i^i  ( A  X.  A
) ) Se  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465   A.wral 2393   {crab 2397   _Vcvv 2660    i^i cin 3040   class class class wbr 3899   Se wse 4221    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-se 4225  df-xp 4515
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator