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Theorem iinrab2 4231
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 4183 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  (/)  { y  e.  B  |  ph } )
2 0iin 4226 . . . . . 6  |-  |^|_ x  e.  (/)  { y  e.  B  |  ph }  =  _V
31, 2syl6eq 2489 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  _V )
43ineq1d 3549 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  ( _V  i^i  B
) )
5 incom 3541 . . . . 5  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
6 inv1 3662 . . . . 5  |-  ( B  i^i  _V )  =  B
75, 6eqtri 2461 . . . 4  |-  ( _V 
i^i  B )  =  B
84, 7syl6eq 2489 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  B )
9 rzal 3779 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ph )
10 rabid2 2896 . . . . 5  |-  ( B  =  { y  e.  B  |  A. x  e.  A  ph }  <->  A. y  e.  B  A. x  e.  A  ph )
11 ralcom 2879 . . . . 5  |-  ( A. y  e.  B  A. x  e.  A  ph  <->  A. x  e.  A  A. y  e.  B  ph )
1210, 11bitr2i 250 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
139, 12sylib 196 . . 3  |-  ( A  =  (/)  ->  B  =  { y  e.  B  |  A. x  e.  A  ph } )
148, 13eqtrd 2473 . 2  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
15 iinrab 4230 . . . 4  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
1615ineq1d 3549 . . 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 3435 . . . 4  |-  { y  e.  B  |  A. x  e.  A  ph }  C_  B
18 dfss 3341 . . . 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 208 . . 3  |-  { y  e.  B  |  A. x  e.  A  ph }  =  ( { y  e.  B  |  A. x  e.  A  ph }  i^i  B )
2016, 19syl6eqr 2491 . 2  |-  ( A  =/=  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
2114, 20pm2.61ine 2685 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 1369    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970    i^i cin 3325    C_ wss 3326   (/)c0 3635   |^|_ciin 4170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3329  df-in 3333  df-ss 3340  df-nul 3636  df-iin 4172
This theorem is referenced by: (None)
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