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Theorem iinrab 4155
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 3725 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2552 . 2  |-  ( A  =/=  (/)  ->  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2716 . . . . 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 4114 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 4154 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2458 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2716 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2495 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 360    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   {crab 2711   (/)c0 3630   |^|_ciin 4096
This theorem is referenced by:  iinrab2  4156  riinrab  4169  ubthlem1  22377  pmapglbx  30639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-nul 3631  df-iin 4098
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