Theorem mptun 5313

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.

Step	Hyp	Ref	Expression
1		df-mpt 4039	. 2 |- ( x e. ( A u. B ) |-> C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
2		df-mpt 4039	. . . 4 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3		df-mpt 4039	. . . 4 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
4	2, 3	uneq12i 3269	. . 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 3258	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 454	. . . . . 6 |- ( ( x e. ( A u. B ) /\ y = C ) <-> ( ( x e. A \/ x e. B ) /\ y = C ) )
7		andir 809	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ y = C ) <-> ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) )
8	6, 7	bitri 183	. . . . 5 |- ( ( x e. ( A u. B ) /\ y = C ) <-> ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) )
9	8	opabbii 4043	. . . 4 |- { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) } = { <. x , y >. | ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) }
10		unopab 4055	. . . 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 ) ) }
11	9, 10	eqtr4i 2188	. . 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 ) } )
12	4, 11	eqtr4i 2188	. 2 |- ( ( x e. A |-> C ) u. ( x e. B |-> C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
13	1, 12	eqtr4i 2188	1 |- ( x e. ( A u. B ) |-> C ) = ( ( x e. A |-> C ) u. ( x e. B |-> C ) )

Syntax hints:   /\ wa 103   \/ wo 698   = wceq 1342   e. wcel 2135   u. cun 3109   {copab 4036   |-> cmpt 4037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-opab 4038  df-mpt 4039

This theorem is referenced by:  fmptap  5669  fmptapd  5670