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Theorem iunab 4325
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 2616 . . . 4  |-  F/_ y A
2 nfab1 2618 . . . 4  |-  F/_ y { y  |  ph }
31, 2nfiun 4307 . . 3  |-  F/_ y U_ x  e.  A  { y  |  ph }
4 nfab1 2618 . . 3  |-  F/_ y { y  |  E. x  e.  A  ph }
53, 4cleqf 2643 . 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 2441 . . . 4  |-  ( y  e.  { y  | 
ph }  <->  ph )
76rexbii 2862 . . 3  |-  ( E. x  e.  A  y  e.  { y  | 
ph }  <->  E. x  e.  A  ph )
8 eliun 4284 . . 3  |-  ( y  e.  U_ x  e.  A  { y  | 
ph }  <->  E. x  e.  A  y  e.  { y  |  ph }
)
9 abid 2441 . . 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 1596 1  |-  U_ x  e.  A  { y  |  ph }  =  {
y  |  E. x  e.  A  ph }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2439   E.wrex 2800   U_ciun 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-iun 4282
This theorem is referenced by:  iunrab  4326  iunid  4334  dfimafn2  5851  rabiun  28561  dfaimafn2  30221  rnfdmpr  30298
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