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Theorem inrab 3453
Description: Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
Assertion
Ref Expression
inrab  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  ps ) }

Proof of Theorem inrab
StepHypRef Expression
1 df-rab 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2565 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2ineq12i 3381 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2565 . . 3  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
5 inab 3449 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
6 anandi 801 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  ps )
)  <->  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps )
) )
76abbii 2408 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2319 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  ps ) ) }
94, 8eqtr4i 2319 . 2  |-  { x  e.  A  |  ( ph  /\  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2319 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    i^i cin 3164
This theorem is referenced by:  rabnc  3491  ixxin  10689  hashbclem  11406  phiprmpw  12860  submacs  14458  ablfacrp  15317  dfrhm2  15514  ordtbaslem  16934  ordtbas2  16937  ordtopn3  16942  ordtcld3  16945  ordthauslem  17127  pthaus  17348  xkohaus  17363  tsmsfbas  17826  minveclem3b  18808  shftmbl  18912  mumul  20435  ppiub  20459  lgsquadlem2  20610  xppreima  23226  xpinpreima  23305  xpinpreima2  23306  hasheuni  23468  measvuni  23557  subfacp1lem6  23731  itg2addnclem2  25004  anrabdioph  26963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-in 3172
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