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Theorem mptun 5695
Description: Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
mptun  |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )

Proof of Theorem mptun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4455 . 2  |-  ( x  e.  ( A  u.  B )  |->  C )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
2 df-mpt 4455 . . . 4  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
3 df-mpt 4455 . . . 4  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
42, 3uneq12i 3595 . . 3  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) } )
5 elun 3584 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 693 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C ) )
7 andir 869 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C )  <->  ( (
x  e.  A  /\  y  =  C )  \/  ( x  e.  B  /\  y  =  C
) ) )
86, 7bitri 249 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) )
98opabbii 4459 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
10 unopab 4470 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
119, 10eqtr4i 2434 . . 3  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  u.  {
<. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) } )
124, 11eqtr4i 2434 . 2  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
131, 12eqtr4i 2434 1  |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    u. cun 3412   {copab 4452    |-> cmpt 4453
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-v 3061  df-un 3419  df-opab 4454  df-mpt 4455
This theorem is referenced by:  fmptap  6074  fmptapd  6075  partfun  27960  esumrnmpt2  28515  ptrest  31420
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