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Theorem mptun 5516
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 4210 . 2  |-  ( x  e.  ( A  u.  B )  |->  C )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
2 df-mpt 4210 . . . 4  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
3 df-mpt 4210 . . . 4  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
42, 3uneq12i 3443 . . 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 3432 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 677 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C ) )
7 andir 839 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C )  <->  ( (
x  e.  A  /\  y  =  C )  \/  ( x  e.  B  /\  y  =  C
) ) )
86, 7bitri 241 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) )
98opabbii 4214 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
10 unopab 4226 . . . 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 2411 . . 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 2411 . 2  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
131, 12eqtr4i 2411 1  |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    u. cun 3262   {copab 4207    e. cmpt 4208
This theorem is referenced by:  fmptap  5863  fmptapd  23904  partfun  23929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-un 3269  df-opab 4209  df-mpt 4210
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