Theorem mptun 5454

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 4146	. 2 |- ( x e. ( A u. B ) |-> C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
2		df-mpt 4146	. . . 4 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3		df-mpt 4146	. . . 4 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
4	2, 3	uneq12i 3356	. . 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 3345	. . . . . . 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 824	. . . . . 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 4150	. . . 4 |- { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) } = { <. x , y >. | ( ( x e. A /\ y = C ) \/ ( x e. B /\ y = C ) ) }
10		unopab 4162	. . . 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 2253	. . 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 2253	. 2 |- ( ( x e. A |-> C ) u. ( x e. B |-> C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y = C ) }
13	1, 12	eqtr4i 2253	1 |- ( x e. ( A u. B ) |-> C ) = ( ( x e. A |-> C ) u. ( x e. B |-> C ) )

Syntax hints:   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   u. cun 3195   {copab 4143   |-> cmpt 4144

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-opab 4145  df-mpt 4146

This theorem is referenced by:  fmptap  5828  fmptapd  5829