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Theorem riinrab 3977
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 3975 . . 3  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  A )
2 rzal 3555 . . . . 5  |-  ( X  =  (/)  ->  A. x  e.  X  ph )
32ralrimivw 2627 . . . 4  |-  ( X  =  (/)  ->  A. y  e.  A  A. x  e.  X  ph )
4 rabid2 2717 . . . 4  |-  ( A  =  { y  e.  A  |  A. x  e.  X  ph }  <->  A. y  e.  A  A. x  e.  X  ph )
53, 4sylibr 203 . . 3  |-  ( X  =  (/)  ->  A  =  { y  e.  A  |  A. x  e.  X  ph } )
61, 5eqtrd 2315 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
7 ssrab2 3258 . . . . 5  |-  { y  e.  A  |  ph }  C_  A
87rgenw 2610 . . . 4  |-  A. x  e.  X  { y  e.  A  |  ph }  C_  A
9 riinn0 3976 . . . 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 651 . . 3  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  = 
|^|_ x  e.  X  { y  e.  A  |  ph } )
11 iinrab 3964 . . 3  |-  ( X  =/=  (/)  ->  |^|_ x  e.  X  { y  e.  A  |  ph }  =  { y  e.  A  |  A. x  e.  X  ph } )
1210, 11eqtrd 2315 . 2  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
136, 12pm2.61ine 2522 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 1623    =/= wne 2446   A.wral 2543   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|_ciin 3906
This theorem is referenced by:  acsfn1  13563  acsfn1c  13564  acsfn2  13565  cntziinsn  14810  csscld  18676  acsfn1p  27507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iin 3908
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