Theorem xpundi 5171

Description: Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
xpundi	|- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )

Proof of Theorem xpundi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xp 5120	. 2 |- ( A X. ( B u. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
2		df-xp 5120	. . . 4 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
3		df-xp 5120	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
4	2, 3	uneq12i 3765	. . 3 |- ( ( A X. B ) u. ( A X. C ) ) = ( { <. x , y >. | ( x e. A /\ y e. B ) } u. { <. x , y >. | ( x e. A /\ y e. C ) } )
5		elun 3753	. . . . . . 7 |- ( y e. ( B u. C ) <-> ( y e. B \/ y e. C ) )
6	5	anbi2i 730	. . . . . 6 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( x e. A /\ ( y e. B \/ y e. C ) ) )
7		andi 911	. . . . . 6 |- ( ( x e. A /\ ( y e. B \/ y e. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
8	6, 7	bitri 264	. . . . 5 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
9	8	opabbii 4717	. . . 4 |- { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) } = { <. x , y >. | ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) }
10		unopab 4728	. . . 4 |- ( { <. x , y >. | ( x e. A /\ y e. B ) } u. { <. x , y >. | ( x e. A /\ y e. C ) } ) = { <. x , y >. | ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) }
11	9, 10	eqtr4i 2647	. . 3 |- { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) } = ( { <. x , y >. | ( x e. A /\ y e. B ) } u. { <. x , y >. | ( x e. A /\ y e. C ) } )
12	4, 11	eqtr4i 2647	. 2 |- ( ( A X. B ) u. ( A X. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
13	1, 12	eqtr4i 2647	1 |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-opab 4713  df-xp 5120

This theorem is referenced by:  xpun  5176  xp2cda  9002  xpcdaen  9005  alephadd  9399  ustund  22025  bj-2upln1upl  33012