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Theorem riinrab 4401
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 4399 . . 3  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  A )
2 rzal 3929 . . . . 5  |-  ( X  =  (/)  ->  A. x  e.  X  ph )
32ralrimivw 2879 . . . 4  |-  ( X  =  (/)  ->  A. y  e.  A  A. x  e.  X  ph )
4 rabid2 3039 . . . 4  |-  ( A  =  { y  e.  A  |  A. x  e.  X  ph }  <->  A. y  e.  A  A. x  e.  X  ph )
53, 4sylibr 212 . . 3  |-  ( X  =  (/)  ->  A  =  { y  e.  A  |  A. x  e.  X  ph } )
61, 5eqtrd 2508 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
7 ssrab2 3585 . . . . 5  |-  { y  e.  A  |  ph }  C_  A
87rgenw 2825 . . . 4  |-  A. x  e.  X  { y  e.  A  |  ph }  C_  A
9 riinn0 4400 . . . 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 670 . . 3  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  = 
|^|_ x  e.  X  { y  e.  A  |  ph } )
11 iinrab 4387 . . 3  |-  ( X  =/=  (/)  ->  |^|_ x  e.  X  { y  e.  A  |  ph }  =  { y  e.  A  |  A. x  e.  X  ph } )
1210, 11eqtrd 2508 . 2  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
136, 12pm2.61ine 2780 1  |-  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    =/= wne 2662   A.wral 2814   {crab 2818    i^i cin 3475    C_ wss 3476   (/)c0 3785   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-iin 4328
This theorem is referenced by:  acsfn1  14912  acsfn1c  14913  acsfn2  14914  cntziinsn  16167  csscld  21424  acsfn1p  30753
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