MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inrab2 Structured version   Unicode version

Theorem inrab2 3614
Description: Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
Assertion
Ref Expression
inrab2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem inrab2
StepHypRef Expression
1 df-rab 2714 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 abid2 2553 . . . 4  |-  { x  |  x  e.  B }  =  B
32eqcomi 2440 . . 3  |-  B  =  { x  |  x  e.  B }
41, 3ineq12i 3540 . 2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
5 df-rab 2714 . . 3  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  { x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
6 inab 3609 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B ) }
7 elin 3530 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
87anbi1i 677 . . . . . 6  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  x  e.  B )  /\  ph ) )
9 an32 774 . . . . . 6  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  /\  x  e.  B ) )
108, 9bitri 241 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) )
1110abbii 2548 . . . 4  |-  { x  |  ( x  e.  ( A  i^i  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) }
126, 11eqtr4i 2459 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
135, 12eqtr4i 2459 . 2  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
144, 13eqtr4i 2459 1  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2709    i^i cin 3319
This theorem is referenced by:  iooval2  10949  fzval2  11046  smuval2  12994  smueqlem  13002  dfphi2  13163  ordtrest  17266  ordtrest2lem  17267  itg2addnclem2  26257
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-in 3327
  Copyright terms: Public domain W3C validator