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Theorem iinrab2 4378
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 4330 . . . . . 6  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  (/)  { y  e.  B  |  ph } )
2 0iin 4373 . . . . . 6  |-  |^|_ x  e.  (/)  { y  e.  B  |  ph }  =  _V
31, 2syl6eq 2511 . . . . 5  |-  ( A  =  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  _V )
43ineq1d 3685 . . . 4  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  ( _V  i^i  B
) )
5 incom 3677 . . . . 5  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
6 inv1 3811 . . . . 5  |-  ( B  i^i  _V )  =  B
75, 6eqtri 2483 . . . 4  |-  ( _V 
i^i  B )  =  B
84, 7syl6eq 2511 . . 3  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  B )
9 rzal 3919 . . . 4  |-  ( A  =  (/)  ->  A. x  e.  A  A. y  e.  B  ph )
10 rabid2 3032 . . . . 5  |-  ( B  =  { y  e.  B  |  A. x  e.  A  ph }  <->  A. y  e.  B  A. x  e.  A  ph )
11 ralcom 3015 . . . . 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 2495 . 2  |-  ( A  =  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
15 iinrab 4377 . . . 4  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  A. x  e.  A  ph } )
1615ineq1d 3685 . . 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 3571 . . . 4  |-  { y  e.  B  |  A. x  e.  A  ph }  C_  B
18 dfss 3476 . . . 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 2513 . 2  |-  ( A  =/=  (/)  ->  ( |^|_ x  e.  A  { y  e.  B  |  ph }  i^i  B )  =  { y  e.  B  |  A. x  e.  A  ph } )
2114, 20pm2.61ine 2767 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 1398    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783   |^|_ciin 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-iin 4318
This theorem is referenced by: (None)
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