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Theorem iinrab 3980
Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinrab  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
Distinct variable groups:    y, A, x    x, B
Allowed substitution hints:    ph( x, y)    B( y)

Proof of Theorem iinrab
StepHypRef Expression
1 r19.28zv 3562 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2410 . 2  |-  ( A  =/=  (/)  ->  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2565 . . . . 5  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
43a1i 10 . . . 4  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
54iineq2i 3940 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 3979 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2316 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2565 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2353 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   {crab 2560   (/)c0 3468   |^|_ciin 3922
This theorem is referenced by:  iinrab2  3981  riinrab  3993  ubthlem1  21465  pmapglbx  30580
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-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-nul 3469  df-iin 3924
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