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Theorem partfun 27188
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 5710 . 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 3905 . . 3  |-  ( ( A  i^i  B )  u.  ( A  \  B ) )  =  A
3 eqid 2467 . . 3  |-  if ( x  e.  B ,  C ,  D )  =  if ( x  e.  B ,  C ,  D )
42, 3mpteq12i 4531 . 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 3719 . . . . . 6  |-  ( A  i^i  B )  C_  B
65sseli 3500 . . . . 5  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  B )
7 iftrue 3945 . . . . 5  |-  ( x  e.  B  ->  if ( x  e.  B ,  C ,  D )  =  C )
86, 7syl 16 . . . 4  |-  ( x  e.  ( A  i^i  B )  ->  if (
x  e.  B ,  C ,  D )  =  C )
98mpteq2ia 4529 . . 3  |-  ( x  e.  ( A  i^i  B )  |->  if ( x  e.  B ,  C ,  D ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
10 eldifn 3627 . . . . 5  |-  ( x  e.  ( A  \  B )  ->  -.  x  e.  B )
11 iffalse 3948 . . . . 5  |-  ( -.  x  e.  B  ->  if ( x  e.  B ,  C ,  D )  =  D )
1210, 11syl 16 . . . 4  |-  ( x  e.  ( A  \  B )  ->  if ( x  e.  B ,  C ,  D )  =  D )
1312mpteq2ia 4529 . . 3  |-  ( x  e.  ( A  \  B )  |->  if ( x  e.  B ,  C ,  D )
)  =  ( x  e.  ( A  \  B )  |->  D )
149, 13uneq12i 3656 . 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 ) )
151, 4, 143eqtr3i 2504 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:   -. wn 3    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475   ifcif 3939    |-> cmpt 4505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-if 3940  df-opab 4506  df-mpt 4507
This theorem is referenced by: (None)
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