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Theorem iinrab2 4332
Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinrab2  |-  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph }
Distinct variable groups:    y, A, x    x, B, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem iinrab2
StepHypRef Expression
1 iineq1 4284 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  (/)  { y  e.  B  |  ph } )
2 0iin 4327 . . . . . 6  |-  |^|_ x  e.  (/)  { y  e.  B  |  ph }  =  _V
31, 2syl6eq 2521 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  _V )
43ineq1d 3624 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  ( _V  i^i  B
) )
5 incom 3616 . . . . 5  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
6 inv1 3764 . . . . 5  |-  ( B  i^i  _V )  =  B
75, 6eqtri 2493 . . . 4  |-  ( _V 
i^i  B )  =  B
84, 7syl6eq 2521 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  B )
9 rzal 3862 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ph )
10 rabid2 2954 . . . . 5  |-  ( B  =  { y  e.  B  |  A. x  e.  A  ph }  <->  A. y  e.  B  A. x  e.  A  ph )
11 ralcom 2937 . . . . 5  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. x  e.  A  A. y  e.  B  ph )
1210, 11bitr2i 258 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
139, 12sylib 201 . . 3  |-  ( A  =  (/)  ->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
148, 13eqtrd 2505 . 2  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
15 iinrab 4331 . . . 4  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
1615ineq1d 3624 . . 3  |-  ( A  =/=  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  ( { y  e.  B  |  A. x  e.  A  ph }  i^i  B ) )
17 ssrab2 3500 . . . 4  |-  { y  e.  B  |  A. x  e.  A  ph }  C_  B
18 dfss 3405 . . . 4  |-  ( { y  e.  B  |  A. x  e.  A  ph }  C_  B  <->  { y  e.  B  |  A. x  e.  A  ph }  =  ( { y  e.  B  |  A. x  e.  A  ph }  i^i  B ) )
1917, 18mpbi 213 . . 3  |-  { y  e.  B  |  A. x  e.  A  ph }  =  ( { y  e.  B  |  A. x  e.  A  ph }  i^i  B )
2016, 19syl6eqr 2523 . 2  |-  ( A  =/=  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
2114, 20pm2.61ine 2726 1  |-  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   |^|_ciin 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-iin 4272
This theorem is referenced by: (None)
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