Theorem partfun 29475

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

Step	Hyp	Ref	Expression
1		mptun 6025	. 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 4046	. . 3 |- ( ( A i^i B ) u. ( A \ B ) ) = A
3		eqid 2622	. . 3 |- if ( x e. B , C , D ) = if ( x e. B , C , D )
4	2, 3	mpteq12i 4742	. 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		elinel2 3800	. . . . 5 |- ( x e. ( A i^i B ) -> x e. B )
6	5	iftrued 4094	. . . 4 |- ( x e. ( A i^i B ) -> if ( x e. B , C , D ) = C )
7	6	mpteq2ia 4740	. . 3 |- ( x e. ( A i^i B ) |-> if ( x e. B , C , D ) ) = ( x e. ( A i^i B ) |-> C )
8		eldifn 3733	. . . . 5 |- ( x e. ( A \ B ) -> -. x e. B )
9	8	iffalsed 4097	. . . 4 |- ( x e. ( A \ B ) -> if ( x e. B , C , D ) = D )
10	9	mpteq2ia 4740	. . 3 |- ( x e. ( A \ B ) |-> if ( x e. B , C , D ) ) = ( x e. ( A \ B ) |-> D )
11	7, 10	uneq12i 3765	. 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 ) )
12	1, 4, 11	3eqtr3i 2652	1 |- ( x e. A |-> if ( x e. B , C , D ) ) = ( ( x e. ( A i^i B ) |-> C ) u. ( x e. ( A \ B ) |-> D ) )

Syntax hints:   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   ifcif 4086   |-> cmpt 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-if 4087  df-opab 4713  df-mpt 4730

This theorem is referenced by: (None)