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Theorem iunab 4317
Description: The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
Assertion
Ref Expression
iunab  |-  U_ x  e.  A  { y  |  ph }  =  {
y  |  E. x  e.  A  ph }
Distinct variable groups:    y, A    x, y
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem iunab
StepHypRef Expression
1 nfcv 2564 . . . 4  |-  F/_ y A
2 nfab1 2566 . . . 4  |-  F/_ y { y  |  ph }
31, 2nfiun 4299 . . 3  |-  F/_ y U_ x  e.  A  { y  |  ph }
4 nfab1 2566 . . 3  |-  F/_ y { y  |  E. x  e.  A  ph }
53, 4cleqf 2591 . 2  |-  ( U_ x  e.  A  {
y  |  ph }  =  { y  |  E. x  e.  A  ph }  <->  A. y ( y  e. 
U_ x  e.  A  { y  |  ph } 
<->  y  e.  { y  |  E. x  e.  A  ph } ) )
6 abid 2389 . . . 4  |-  ( y  e.  { y  | 
ph }  <->  ph )
76rexbii 2906 . . 3  |-  ( E. x  e.  A  y  e.  { y  | 
ph }  <->  E. x  e.  A  ph )
8 eliun 4276 . . 3  |-  ( y  e.  U_ x  e.  A  { y  | 
ph }  <->  E. x  e.  A  y  e.  { y  |  ph }
)
9 abid 2389 . . 3  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
107, 8, 93bitr4i 277 . 2  |-  ( y  e.  U_ x  e.  A  { y  | 
ph }  <->  y  e.  { y  |  E. x  e.  A  ph } )
115, 10mpgbir 1643 1  |-  U_ x  e.  A  { y  |  ph }  =  {
y  |  E. x  e.  A  ph }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2755   U_ciun 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-v 3061  df-iun 4273
This theorem is referenced by:  iunrab  4318  iunid  4326  dfimafn2  5899  rabiun  31408  dfaimafn2  37619  rnfdmpr  37941
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