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Theorem inrab2 3620
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 2722 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 abid2 2558 . . . 4  |-  { x  |  x  e.  B }  =  B
32eqcomi 2445 . . 3  |-  B  =  { x  |  x  e.  B }
41, 3ineq12i 3547 . 2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
5 df-rab 2722 . . 3  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  { x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
6 inab 3615 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B ) }
7 elin 3536 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
87anbi1i 690 . . . . . 6  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  x  e.  B )  /\  ph ) )
9 an32 791 . . . . . 6  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  /\  x  e.  B ) )
108, 9bitri 249 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) )
1110abbii 2553 . . . 4  |-  { x  |  ( x  e.  ( A  i^i  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) }
126, 11eqtr4i 2464 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
135, 12eqtr4i 2464 . 2  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
144, 13eqtr4i 2464 1  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1364    e. wcel 1761   {cab 2427   {crab 2717    i^i cin 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-in 3332
This theorem is referenced by:  iooval2  11329  fzval2  11436  smuval2  13674  smueqlem  13682  dfphi2  13845  ordtrest  18765  ordtrest2lem  18766  ordtrestNEW  26287  ordtrest2NEWlem  26288  itg2addnclem2  28369
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