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Theorem mptun 5665
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 4422 . 2  |-  ( x  e.  ( A  u.  B )  |->  C )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
2 df-mpt 4422 . . . 4  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
3 df-mpt 4422 . . . 4  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
42, 3uneq12i 3556 . . 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 3544 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 699 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C ) )
7 andir 876 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C )  <->  ( (
x  e.  A  /\  y  =  C )  \/  ( x  e.  B  /\  y  =  C
) ) )
86, 7bitri 252 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) )
98opabbii 4426 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
10 unopab 4437 . . . 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 2448 . . 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 2448 . 2  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
131, 12eqtr4i 2448 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 369    /\ wa 370    = wceq 1437    e. wcel 1872    u. cun 3372   {copab 4419    |-> cmpt 4420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-v 3019  df-un 3379  df-opab 4421  df-mpt 4422
This theorem is referenced by:  fmptap  6041  fmptapd  6042  partfun  28219  esumrnmpt2  28836  ptrest  31846
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