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Theorem iinrabm 3882
Description: Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iinrabm  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  { y  e.  B  |  ph }  =  {
y  e.  B  |  A. x  e.  A  ph } )
Distinct variable groups:    y, A, x   
x, B
Allowed substitution hints:    ph( x, y)    B( y)

Proof of Theorem iinrabm
StepHypRef Expression
1 r19.28mv 3459 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  A. x  e.  A  ph ) ) )
21abbidv 2258 . 2  |-  ( E. x  x  e.  A  ->  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }  =  {
y  |  ( y  e.  B  /\  A. x  e.  A  ph ) } )
3 df-rab 2426 . . . . 5  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
43a1i 9 . . . 4  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
54iineq2i 3839 . . 3  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
6 iinab 3881 . . 3  |-  |^|_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  A. x  e.  A  ( y  e.  B  /\  ph ) }
75, 6eqtri 2161 . 2  |-  |^|_ x  e.  A  { y  e.  B  |  ph }  =  { y  |  A. x  e.  A  (
y  e.  B  /\  ph ) }
8 df-rab 2426 . 2  |-  { y  e.  B  |  A. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  A. x  e.  A  ph ) }
92, 7, 83eqtr4g 2198 1  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  { y  e.  B  |  ph }  =  {
y  e.  B  |  A. x  e.  A  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   {cab 2126   A.wral 2417   {crab 2421   |^|_ciin 3821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-iin 3823
This theorem is referenced by: (None)
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