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Theorem iunab 3798
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 2235 . . . 4  |-  F/_ y A
2 nfab1 2237 . . . 4  |-  F/_ y { y  |  ph }
31, 2nfiunxy 3778 . . 3  |-  F/_ y U_ x  e.  A  { y  |  ph }
4 nfab1 2237 . . 3  |-  F/_ y { y  |  E. x  e.  A  ph }
53, 4cleqf 2259 . 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 2083 . . . 4  |-  ( y  e.  { y  | 
ph }  <->  ph )
76rexbii 2396 . . 3  |-  ( E. x  e.  A  y  e.  { y  | 
ph }  <->  E. x  e.  A  ph )
8 eliun 3756 . . 3  |-  ( y  e.  U_ x  e.  A  { y  | 
ph }  <->  E. x  e.  A  y  e.  { y  |  ph }
)
9 abid 2083 . . 3  |-  ( y  e.  { y  |  E. x  e.  A  ph }  <->  E. x  e.  A  ph )
107, 8, 93bitr4i 211 . 2  |-  ( y  e.  U_ x  e.  A  { y  | 
ph }  <->  y  e.  { y  |  E. x  e.  A  ph } )
115, 10mpgbir 1394 1  |-  U_ x  e.  A  { y  |  ph }  =  {
y  |  E. x  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1296    e. wcel 1445   {cab 2081   E.wrex 2371   U_ciun 3752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-iun 3754
This theorem is referenced by:  iunrab  3799  iunid  3807  dfimafn2  5389
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