Theorem rabun2 3386

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 2444	. 2 |- { x e. ( A u. B ) | ph } = { x | ( x e. ( A u. B ) /\ ph ) }
2		df-rab 2444	. . . 4 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3		df-rab 2444	. . . 4 |- { x e. B | ph } = { x | ( x e. B /\ ph ) }
4	2, 3	uneq12i 3259	. . 3 |- ( { x e. A | ph } u. { x e. B | ph } ) = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
5		elun 3248	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 454	. . . . . 6 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A \/ x e. B ) /\ ph ) )
7		andir 809	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
8	6, 7	bitri 183	. . . . 5 |- ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
9	8	abbii 2273	. . . 4 |- { x | ( x e. ( A u. B ) /\ ph ) } = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) }
10		unab 3374	. . . 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 2181	. . 3 |- { x | ( x e. ( A u. B ) /\ ph ) } = ( { x | ( x e. A /\ ph ) } u. { x | ( x e. B /\ ph ) } )
12	4, 11	eqtr4i 2181	. 2 |- ( { x e. A | ph } u. { x e. B | ph } ) = { x | ( x e. ( A u. B ) /\ ph ) }
13	1, 12	eqtr4i 2181	1 |- { x e. ( A u. B ) | ph } = ( { x e. A | ph } u. { x e. B | ph } )

Syntax hints:   /\ wa 103   \/ wo 698   = wceq 1335   e. wcel 2128   {cab 2143   {crab 2439   u. cun 3100

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106

This theorem is referenced by:  ssfirab  6878