MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinrab Structured version   Unicode version

Theorem iinrab 4145
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 3715 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2549 . 2  |-  ( A  =/=  (/)  ->  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2706 . . . . 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 4104 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 4144 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2455 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2706 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2492 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 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   {crab 2701   (/)c0 3620   |^|_ciin 4086
This theorem is referenced by:  iinrab2  4146  riinrab  4158  ubthlem1  22364  pmapglbx  30503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-nul 3621  df-iin 4088
  Copyright terms: Public domain W3C validator