Theorem rabun2 3483

Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Assertion

Ref	Expression
rabun2	|- { x e. ( A u. B ) | ph } = ( { x e. A | ph } u. { x e. B | ph } )

Proof of Theorem rabun2

Step	Hyp	Ref	Expression
1		df-rab 2517	. 2 |- { x e. ( A u. B ) | ph } = { x | ( x e. ( A u. B ) /\ ph ) }
2		df-rab 2517	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2517	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
4	2, 3	uneq12i 3356	. . 3 |- ( { x e. A | ph } u. { x e. B | ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
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 ) /\ ph ) <-> ( ( x e. A \/ x e. B ) /\ ph ) )
7		andir 824	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
8	6, 7	bitri 184	. . . . 5 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
9	8	abbii 2345	. . . 4 |- { x | ( x e. ( A u. B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) }
10		unab 3471	. . . 4 |- ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) }
11	9, 10	eqtr4i 2253	. . 3 |- { x | ( x e. ( A u. B ) /\ ph ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
12	4, 11	eqtr4i 2253	. 2 |- ( { x e. A | ph } u. { x e. B | ph } ) = { x | ( x e. ( A u. B ) /\ ph ) }
13	1, 12	eqtr4i 2253	1 |- { x e. ( A u. B ) | ph } = ( { x e. A | ph } u. { x e. B | ph } )

Syntax hints:   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   u. cun 3195

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-rab 2517  df-v 2801  df-un 3201

This theorem is referenced by:  ssfirab  7098