MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunrab Unicode version

Theorem iunrab 3965
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 3964 . 2  |-  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }  =  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }
2 df-rab 2565 . . . 4  |-  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) }
32a1i 10 . . 3  |-  ( x  e.  A  ->  { y  e.  B  |  ph }  =  { y  |  ( y  e.  B  /\  ph ) } )
43iuneq2i 3939 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  =  U_ x  e.  A  { y  |  ( y  e.  B  /\  ph ) }
5 df-rab 2565 . . 3  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
6 r19.42v 2707 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  ph )  <->  ( y  e.  B  /\  E. x  e.  A  ph ) )
76abbii 2408 . . 3  |-  { y  |  E. x  e.  A  ( y  e.  B  /\  ph ) }  =  { y  |  ( y  e.  B  /\  E. x  e.  A  ph ) }
85, 7eqtr4i 2319 . 2  |-  { y  e.  B  |  E. x  e.  A  ph }  =  { y  |  E. x  e.  A  (
y  e.  B  /\  ph ) }
91, 4, 83eqtr4i 2326 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 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   {crab 2560   U_ciun 3921
This theorem is referenced by:  incexc2  12313  itg2monolem1  19121  aannenlem1  19724  musum  20447  lgsquadlem1  20609  lgsquadlem2  20610  fiphp3d  27005  phisum  27621  mapdval3N  32443  mapdval5N  32445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-iun 3923
  Copyright terms: Public domain W3C validator