ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mptun Unicode version

Theorem mptun 5109
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 3876 . 2  |-  ( x  e.  ( A  u.  B )  |->  C )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
2 df-mpt 3876 . . . 4  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
3 df-mpt 3876 . . . 4  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
42, 3uneq12i 3141 . . 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 3130 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 446 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C ) )
7 andir 766 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  y  =  C )  <->  ( (
x  e.  A  /\  y  =  C )  \/  ( x  e.  B  /\  y  =  C
) ) )
86, 7bitri 182 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  y  =  C )  <->  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) )
98opabbii 3880 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  C
)  \/  ( x  e.  B  /\  y  =  C ) ) }
10 unopab 3892 . . . 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 2108 . . 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 2108 . 2  |-  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  u.  B )  /\  y  =  C ) }
131, 12eqtr4i 2108 1  |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662    = wceq 1287    e. wcel 1436    u. cun 2986   {copab 3873    |-> cmpt 3874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-opab 3875  df-mpt 3876
This theorem is referenced by:  fmptap  5450  fmptapd  5451
  Copyright terms: Public domain W3C validator