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Theorem iinrab2 3965
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 3919 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  (/)  { y  e.  B  |  ph } )
2 0iin 3960 . . . . . 6  |-  |^|_ x  e.  (/)  { y  e.  B  |  ph }  =  _V
31, 2syl6eq 2331 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  _V )
43ineq1d 3369 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  ( _V  i^i  B
) )
5 incom 3361 . . . . 5  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
6 inv1 3481 . . . . 5  |-  ( B  i^i  _V )  =  B
75, 6eqtri 2303 . . . 4  |-  ( _V 
i^i  B )  =  B
84, 7syl6eq 2331 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  B )
9 rzal 3555 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ph )
10 rabid2 2717 . . . . 5  |-  ( B  =  { y  e.  B  |  A. x  e.  A  ph }  <->  A. y  e.  B  A. x  e.  A  ph )
11 ralcom 2700 . . . . 5  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. x  e.  A  A. y  e.  B  ph )
1210, 11bitr2i 241 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
139, 12sylib 188 . . 3  |-  ( A  =  (/)  ->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
148, 13eqtrd 2315 . 2  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
15 iinrab 3964 . . . 4  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
1615ineq1d 3369 . . 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 3258 . . . 4  |-  { y  e.  B  |  A. x  e.  A  ph }  C_  B
18 dfss 3167 . . . 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 199 . . 3  |-  { y  e.  B  |  A. x  e.  A  ph }  =  ( { y  e.  B  |  A. x  e.  A  ph }  i^i  B )
2016, 19syl6eqr 2333 . 2  |-  ( A  =/=  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
2114, 20pm2.61ine 2522 1  |-  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|_ciin 3906
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-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iin 3908
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