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Theorem riinrab 4107
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 4105 . . 3  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  A )
2 rzal 3672 . . . . 5  |-  ( X  =  (/)  ->  A. x  e.  X  ph )
32ralrimivw 2733 . . . 4  |-  ( X  =  (/)  ->  A. y  e.  A  A. x  e.  X  ph )
4 rabid2 2828 . . . 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 2419 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
7 ssrab2 3371 . . . . 5  |-  { y  e.  A  |  ph }  C_  A
87rgenw 2716 . . . 4  |-  A. x  e.  X  { y  e.  A  |  ph }  C_  A
9 riinn0 4106 . . . 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 4094 . . 3  |-  ( X  =/=  (/)  ->  |^|_ x  e.  X  { y  e.  A  |  ph }  =  { y  e.  A  |  A. x  e.  X  ph } )
1210, 11eqtrd 2419 . 2  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
136, 12pm2.61ine 2626 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 1649    =/= wne 2550   A.wral 2649   {crab 2653    i^i cin 3262    C_ wss 3263   (/)c0 3571   |^|_ciin 4036
This theorem is referenced by:  acsfn1  13813  acsfn1c  13814  acsfn2  13815  cntziinsn  15060  csscld  19074  acsfn1p  27176
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-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-nul 3572  df-iin 4038
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