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Theorem inrab2 3454
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 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 abid2 2413 . . . 4  |-  { x  |  x  e.  B }  =  B
32eqcomi 2300 . . 3  |-  B  =  { x  |  x  e.  B }
41, 3ineq12i 3381 . 2  |-  ( { x  e.  A  |  ph }  i^i  B )  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
5 df-rab 2565 . . 3  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  { x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
6 inab 3449 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B ) }
7 elin 3371 . . . . . . 7  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
87anbi1i 676 . . . . . 6  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  x  e.  B )  /\  ph ) )
9 an32 773 . . . . . 6  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  /\  x  e.  B ) )
108, 9bitri 240 . . . . 5  |-  ( ( x  e.  ( A  i^i  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) )
1110abbii 2408 . . . 4  |-  { x  |  ( x  e.  ( A  i^i  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  x  e.  B
) }
126, 11eqtr4i 2319 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B } )  =  {
x  |  ( x  e.  ( A  i^i  B )  /\  ph ) }
135, 12eqtr4i 2319 . 2  |-  { x  e.  ( A  i^i  B
)  |  ph }  =  ( { x  |  ( x  e.  A  /\  ph ) }  i^i  { x  |  x  e.  B }
)
144, 13eqtr4i 2319 1  |-  ( { x  e.  A  |  ph }  i^i  B )  =  { x  e.  ( A  i^i  B
)  |  ph }
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:  iooval2  10705  fzval2  10801  smuval2  12689  smueqlem  12697  dfphi2  12858  ordtrest  16948  ordtrest2lem  16949  itg2addnclem2  25004  bsstrs  26249  nbssntrs  26250
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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