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Theorem iunrab 4072
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 4071 . 2  |-  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }
2 df-rab 2651 . . . 4  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
32a1i 11 . . 3  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
43iuneq2i 4046 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
5 df-rab 2651 . . 3  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
6 r19.42v 2798 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  E. x  e.  A  ph ) )
76abbii 2492 . . 3  |-  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
85, 7eqtr4i 2403 . 2  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  E. x  e.  A  (
y  e.  B  /\  ph ) }
91, 4, 83eqtr4i 2410 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 359    = wceq 1649    e. wcel 1717   {cab 2366   E.wrex 2643   {crab 2646   U_ciun 4028
This theorem is referenced by:  incexc2  12538  itg2monolem1  19502  aannenlem1  20105  musum  20836  lgsquadlem1  20998  lgsquadlem2  20999  fiphp3d  26564  phisum  27180  mapdval3N  31797  mapdval5N  31799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-in 3263  df-ss 3270  df-iun 4030
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