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Theorem iunopab 4296
Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
iunopab  |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
Distinct variable groups:    x, A    y, A    y, z    x, z
Allowed substitution hints:    ph( x, y, z)    A( z)

Proof of Theorem iunopab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elopab 4272 . . . . 5  |-  ( w  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( w  =  <. x ,  y >.  /\  ph ) )
21rexbii 2568 . . . 4  |-  ( E. z  e.  A  w  e.  { <. x ,  y >.  |  ph } 
<->  E. z  e.  A  E. x E. y ( w  =  <. x ,  y >.  /\  ph ) )
3 rexcom4 2807 . . . . 5  |-  ( E. z  e.  A  E. x E. y ( w  =  <. x ,  y
>.  /\  ph )  <->  E. x E. z  e.  A  E. y ( w  = 
<. x ,  y >.  /\  ph ) )
4 rexcom4 2807 . . . . . . 7  |-  ( E. z  e.  A  E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y E. z  e.  A  ( w  =  <. x ,  y >.  /\  ph ) )
5 r19.42v 2694 . . . . . . . 8  |-  ( E. z  e.  A  ( w  =  <. x ,  y >.  /\  ph ) 
<->  ( w  =  <. x ,  y >.  /\  E. z  e.  A  ph )
)
65exbii 1569 . . . . . . 7  |-  ( E. y E. z  e.  A  ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. x ,  y >.  /\  E. z  e.  A  ph )
)
74, 6bitri 240 . . . . . 6  |-  ( E. z  e.  A  E. y ( w  = 
<. x ,  y >.  /\  ph )  <->  E. y
( w  =  <. x ,  y >.  /\  E. z  e.  A  ph )
)
87exbii 1569 . . . . 5  |-  ( E. x E. z  e.  A  E. y ( w  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( w  =  <. x ,  y >.  /\  E. z  e.  A  ph )
)
93, 8bitri 240 . . . 4  |-  ( E. z  e.  A  E. x E. y ( w  =  <. x ,  y
>.  /\  ph )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z  e.  A  ph ) )
102, 9bitri 240 . . 3  |-  ( E. z  e.  A  w  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( w  =  <. x ,  y >.  /\  E. z  e.  A  ph )
)
1110abbii 2395 . 2  |-  { w  |  E. z  e.  A  w  e.  { <. x ,  y >.  |  ph } }  =  {
w  |  E. x E. y ( w  = 
<. x ,  y >.  /\  E. z  e.  A  ph ) }
12 df-iun 3907 . 2  |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { w  |  E. z  e.  A  w  e.  { <. x ,  y >.  |  ph } }
13 df-opab 4078 . 2  |-  { <. x ,  y >.  |  E. z  e.  A  ph }  =  { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  E. z  e.  A  ph ) }
1411, 12, 133eqtr4i 2313 1  |-  U_ z  e.  A  { <. x ,  y >.  |  ph }  =  { <. x ,  y >.  |  E. z  e.  A  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   <.cop 3643   U_ciun 3905   {copab 4076
This theorem is referenced by:  marypha2lem2  7189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078
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