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Theorem riinrab 4158
Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinrab  |-  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph }
Distinct variable groups:    x, A, y    x, X, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem riinrab
StepHypRef Expression
1 riin0 4156 . . 3  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  A )
2 rzal 3721 . . . . 5  |-  ( X  =  (/)  ->  A. x  e.  X  ph )
32ralrimivw 2782 . . . 4  |-  ( X  =  (/)  ->  A. y  e.  A  A. x  e.  X  ph )
4 rabid2 2877 . . . 4  |-  ( A  =  { y  e.  A  |  A. x  e.  X  ph }  <->  A. y  e.  A  A. x  e.  X  ph )
53, 4sylibr 204 . . 3  |-  ( X  =  (/)  ->  A  =  { y  e.  A  |  A. x  e.  X  ph } )
61, 5eqtrd 2467 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
7 ssrab2 3420 . . . . 5  |-  { y  e.  A  |  ph }  C_  A
87rgenw 2765 . . . 4  |-  A. x  e.  X  { y  e.  A  |  ph }  C_  A
9 riinn0 4157 . . . 4  |-  ( ( A. x  e.  X  { y  e.  A  |  ph }  C_  A  /\  X  =/=  (/) )  -> 
( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  |^|_ x  e.  X  { y  e.  A  |  ph } )
108, 9mpan 652 . . 3  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  = 
|^|_ x  e.  X  { y  e.  A  |  ph } )
11 iinrab 4145 . . 3  |-  ( X  =/=  (/)  ->  |^|_ x  e.  X  { y  e.  A  |  ph }  =  { y  e.  A  |  A. x  e.  X  ph } )
1210, 11eqtrd 2467 . 2  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
136, 12pm2.61ine 2674 1  |-  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1652    =/= wne 2598   A.wral 2697   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   |^|_ciin 4086
This theorem is referenced by:  acsfn1  13878  acsfn1c  13879  acsfn2  13880  cntziinsn  15125  csscld  19195  acsfn1p  27465
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-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-iin 4088
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