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Theorem iunrab 4130
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 4129 . 2  |-  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }
2 df-rab 2706 . . . 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 4103 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
5 df-rab 2706 . . 3  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
6 r19.42v 2854 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  E. x  e.  A  ph ) )
76abbii 2547 . . 3  |-  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
85, 7eqtr4i 2458 . 2  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  E. x  e.  A  (
y  e.  B  /\  ph ) }
91, 4, 83eqtr4i 2465 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698   {crab 2701   U_ciun 4085
This theorem is referenced by:  incexc2  12610  itg2monolem1  19634  aannenlem1  20237  musum  20968  lgsquadlem1  21130  lgsquadlem2  21131  cnambfre  26245  fiphp3d  26871  phisum  27486  mapdval3N  32366  mapdval5N  32368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087
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