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Theorem iunrab 3751
Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
iunrab  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  E. x  e.  A  ph }
Distinct variable groups:    y, A    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem iunrab
StepHypRef Expression
1 iunab 3750 . 2  |-  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }
2 df-rab 2362 . . . 4  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
32a1i 9 . . 3  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
43iuneq2i 3722 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
5 df-rab 2362 . . 3  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
6 r19.42v 2517 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  E. x  e.  A  ph ) )
76abbii 2198 . . 3  |-  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
85, 7eqtr4i 2106 . 2  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  E. x  e.  A  (
y  e.  B  /\  ph ) }
91, 4, 83eqtr4i 2113 1  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  { y  e.  B  |  E. x  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2069   E.wrex 2354   {crab 2357   U_ciun 3704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997  df-iun 3706
This theorem is referenced by: (None)
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