Theorem rabun2 3486

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 2519	. 2 |- { x e. ( A u. B ) | ph } = { x | ( x e. ( A u. B ) /\ ph ) }
2		df-rab 2519	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2519	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
4	2, 3	uneq12i 3359	. . 3 |- ( { x e. A | ph } u. { x e. B | ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
5		elun 3348	. . . . . . 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 826	. . . . . 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 2347	. . . 4 |- { x | ( x e. ( A u. B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) }
10		unab 3474	. . . 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 2255	. . 3 |- { x | ( x e. ( A u. B ) /\ ph ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
12	4, 11	eqtr4i 2255	. 2 |- ( { x e. A | ph } u. { x e. B | ph } ) = { x | ( x e. ( A u. B ) /\ ph ) }
13	1, 12	eqtr4i 2255	1 |- { x e. ( A u. B ) | ph } = ( { x e. A | ph } u. { x e. B | ph } )

Syntax hints:   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   u. cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204

This theorem is referenced by:  ssfirab  7128