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Theorem partfun 27663
Description: Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
Assertion
Ref Expression
partfun  |-  ( x  e.  A  |->  if ( x  e.  B ,  C ,  D )
)  =  ( ( x  e.  ( A  i^i  B )  |->  C )  u.  ( x  e.  ( A  \  B )  |->  D ) )

Proof of Theorem partfun
StepHypRef Expression
1 mptun 5620 . 2  |-  ( x  e.  ( ( A  i^i  B )  u.  ( A  \  B
) )  |->  if ( x  e.  B ,  C ,  D )
)  =  ( ( x  e.  ( A  i^i  B )  |->  if ( x  e.  B ,  C ,  D ) )  u.  ( x  e.  ( A  \  B )  |->  if ( x  e.  B ,  C ,  D )
) )
2 inundif 3822 . . 3  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
3 eqid 2382 . . 3  |-  if ( x  e.  B ,  C ,  D )  =  if ( x  e.  B ,  C ,  D )
42, 3mpteq12i 4451 . 2  |-  ( x  e.  ( ( A  i^i  B )  u.  ( A  \  B
) )  |->  if ( x  e.  B ,  C ,  D )
)  =  ( x  e.  A  |->  if ( x  e.  B ,  C ,  D )
)
5 inss2 3633 . . . . . 6  |-  ( A  i^i  B )  C_  B
65sseli 3413 . . . . 5  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  B )
76iftrued 3865 . . . 4  |-  ( x  e.  ( A  i^i  B )  ->  if (
x  e.  B ,  C ,  D )  =  C )
87mpteq2ia 4449 . . 3  |-  ( x  e.  ( A  i^i  B )  |->  if ( x  e.  B ,  C ,  D ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
9 eldifn 3541 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  -.  x  e.  B )
109iffalsed 3868 . . . 4  |-  ( x  e.  ( A  \  B )  ->  if ( x  e.  B ,  C ,  D )  =  D )
1110mpteq2ia 4449 . . 3  |-  ( x  e.  ( A  \  B )  |->  if ( x  e.  B ,  C ,  D )
)  =  ( x  e.  ( A  \  B )  |->  D )
128, 11uneq12i 3570 . 2  |-  ( ( x  e.  ( A  i^i  B )  |->  if ( x  e.  B ,  C ,  D ) )  u.  ( x  e.  ( A  \  B )  |->  if ( x  e.  B ,  C ,  D )
) )  =  ( ( x  e.  ( A  i^i  B ) 
|->  C )  u.  (
x  e.  ( A 
\  B )  |->  D ) )
131, 4, 123eqtr3i 2419 1  |-  ( x  e.  A  |->  if ( x  e.  B ,  C ,  D )
)  =  ( ( x  e.  ( A  i^i  B )  |->  C )  u.  ( x  e.  ( A  \  B )  |->  D ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    e. wcel 1826    \ cdif 3386    u. cun 3387    i^i cin 3388   ifcif 3857    |-> cmpt 4425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-if 3858  df-opab 4426  df-mpt 4427
This theorem is referenced by: (None)
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