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Theorem iinrab 4094
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 3666 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2501 . 2  |-  ( A  =/=  (/)  ->  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2658 . . . . 5  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
43a1i 11 . . . 4  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
54iineq2i 4054 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 4093 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2407 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2658 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2444 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 359    = wceq 1649    e. wcel 1717   {cab 2373    =/= wne 2550   A.wral 2649   {crab 2653   (/)c0 3571   |^|_ciin 4036
This theorem is referenced by:  iinrab2  4095  riinrab  4107  ubthlem1  22220  pmapglbx  29883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rab 2658  df-v 2901  df-dif 3266  df-nul 3572  df-iin 4038
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