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Theorem partfun 28274
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 5725 . 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 3874 . . 3  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
3 eqid 2423 . . 3  |-  if ( x  e.  B ,  C ,  D )  =  if ( x  e.  B ,  C ,  D )
42, 3mpteq12i 4506 . 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 3684 . . . . . 6  |-  ( A  i^i  B )  C_  B
65sseli 3461 . . . . 5  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  B )
76iftrued 3918 . . . 4  |-  ( x  e.  ( A  i^i  B )  ->  if (
x  e.  B ,  C ,  D )  =  C )
87mpteq2ia 4504 . . 3  |-  ( x  e.  ( A  i^i  B )  |->  if ( x  e.  B ,  C ,  D ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
9 eldifn 3589 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  -.  x  e.  B )
109iffalsed 3921 . . . 4  |-  ( x  e.  ( A  \  B )  ->  if ( x  e.  B ,  C ,  D )  =  D )
1110mpteq2ia 4504 . . 3  |-  ( x  e.  ( A  \  B )  |->  if ( x  e.  B ,  C ,  D )
)  =  ( x  e.  ( A  \  B )  |->  D )
128, 11uneq12i 3619 . 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 2460 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 1438    e. wcel 1869    \ cdif 3434    u. cun 3435    i^i cin 3436   ifcif 3910    |-> cmpt 4480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-if 3911  df-opab 4481  df-mpt 4482
This theorem is referenced by: (None)
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