Theorem mptun 5339

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 4061	. 2 |- ( x e. ( A u. B ) |-> C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
2		df-mpt 4061	. . . 4 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3		df-mpt 4061	. . . 4 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
4	2, 3	uneq12i 3285	. . 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 3274	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 458	. . . . . 6 |- ( ( x e. ( A u. B ) /\ y = C ) <-> ( ( x e. A \/ x e. B ) /\ y = C ) )
7		andir 819	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ y = C ) <-> ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) )
8	6, 7	bitri 184	. . . . 5 |- ( ( x e. ( A u. B ) /\ y = C ) <-> ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) )
9	8	opabbii 4065	. . . 4 |- { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) } = { <. x , y >. | ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) }
10		unopab 4077	. . . 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 2199	. . 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 2199	. 2 |- ( ( x e. A |-> C ) u. ( x e. B |-> C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
13	1, 12	eqtr4i 2199	1 |- ( x e. ( A u. B ) |-> C ) = ( ( x e. A |-> C ) u. ( x e. B |-> C ) )

Syntax hints:   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2146   u. cun 3125   {copab 4058   |-> cmpt 4059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-opab 4060  df-mpt 4061

This theorem is referenced by:  fmptap  5698  fmptapd  5699