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

Theorem riinrab 4346
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 4344 . . 3  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  A )
2 rzal 3874 . . . . 5  |-  ( X  =  (/)  ->  A. x  e.  X  ph )
32ralrimivw 2818 . . . 4  |-  ( X  =  (/)  ->  A. y  e.  A  A. x  e.  X  ph )
4 rabid2 2984 . . . 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 2443 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
7 ssrab2 3523 . . . . 5  |-  { y  e.  A  |  ph }  C_  A
87rgenw 2764 . . . 4  |-  A. x  e.  X  { y  e.  A  |  ph }  C_  A
9 riinn0 4345 . . . 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 668 . . 3  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  = 
|^|_ x  e.  X  { y  e.  A  |  ph } )
11 iinrab 4332 . . 3  |-  ( X  =/=  (/)  ->  |^|_ x  e.  X  { y  e.  A  |  ph }  =  { y  e.  A  |  A. x  e.  X  ph } )
1210, 11eqtrd 2443 . 2  |-  ( X  =/=  (/)  ->  ( A  i^i  |^|_ x  e.  X  { y  e.  A  |  ph } )  =  { y  e.  A  |  A. x  e.  X  ph } )
136, 12pm2.61ine 2716 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 1405    =/= wne 2598   A.wral 2753   {crab 2757    i^i cin 3412    C_ wss 3413   (/)c0 3737   |^|_ciin 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-in 3420  df-ss 3427  df-nul 3738  df-iin 4273
This theorem is referenced by:  acsfn1  15167  acsfn1c  15168  acsfn2  15169  cntziinsn  16588  csscld  21873  acsfn1p  35493
  Copyright terms: Public domain W3C validator