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

Theorem unopab 4514
Description: Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
unopab  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  \/  ps ) }

Proof of Theorem unopab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 unab 3762 . . 3  |-  ( { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )  =  { z  |  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) ) }
2 19.43 1698 . . . . 5  |-  ( E. x ( E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. y
( z  =  <. x ,  y >.  /\  ps ) )  <->  ( E. x E. y ( z  =  <. x ,  y
>.  /\  ph )  \/ 
E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
3 andi 865 . . . . . . . 8  |-  ( ( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) )  <-> 
( ( z  = 
<. x ,  y >.  /\  ph )  \/  (
z  =  <. x ,  y >.  /\  ps ) ) )
43exbii 1672 . . . . . . 7  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) )  <->  E. y
( ( z  = 
<. x ,  y >.  /\  ph )  \/  (
z  =  <. x ,  y >.  /\  ps ) ) )
5 19.43 1698 . . . . . . 7  |-  ( E. y ( ( z  =  <. x ,  y
>.  /\  ph )  \/  ( z  =  <. x ,  y >.  /\  ps ) )  <->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. y ( z  = 
<. x ,  y >.  /\  ps ) ) )
64, 5bitr2i 250 . . . . . 6  |-  ( ( E. y ( z  =  <. x ,  y
>.  /\  ph )  \/ 
E. y ( z  =  <. x ,  y
>.  /\  ps ) )  <->  E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
76exbii 1672 . . . . 5  |-  ( E. x ( E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. y
( z  =  <. x ,  y >.  /\  ps ) )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
82, 7bitr3i 251 . . . 4  |-  ( ( E. x E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
98abbii 2588 . . 3  |-  { z  |  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) ) }  =  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) }
101, 9eqtri 2483 . 2  |-  ( { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) ) }
11 df-opab 4498 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
12 df-opab 4498 . . 3  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
1311, 12uneq12i 3642 . 2  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  ( { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )
14 df-opab 4498 . 2  |-  { <. x ,  y >.  |  (
ph  \/  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) ) }
1510, 13, 143eqtr4i 2493 1  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  \/  ps ) }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    = wceq 1398   E.wex 1617   {cab 2439    u. cun 3459   <.cop 4022   {copab 4496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-opab 4498
This theorem is referenced by:  xpundi  5041  xpundir  5042  cnvun  5396  coundi  5491  coundir  5492  mptun  5694  opsrtoslem1  18346  lgsquadlem3  23832
  Copyright terms: Public domain W3C validator