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Theorem mptun 5652
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 4463 . 2  |-  ( x  e.  ( A  u.  B )  |->  C )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
2 df-mpt 4463 . . . 4  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
3 df-mpt 4463 . . . 4  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
42, 3uneq12i 3619 . . 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 3608 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 695 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C ) )
7 andir 863 . . . . . 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 4467 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
10 unopab 4478 . . . 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 2486 . . 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 2486 . 2  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
131, 12eqtr4i 2486 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 368    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3437   {copab 4460    |-> cmpt 4461
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-v 3080  df-un 3444  df-opab 4462  df-mpt 4463
This theorem is referenced by:  fmptap  6013  fmptapd  6014  partfun  26171  ptrest  28596
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