Theorem xpundir 5042

Description: Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem xpundir

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

Step	Hyp	Ref	Expression
1		df-xp 4994	. 2 |- ( ( A u. B ) X. C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
2		df-xp 4994	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
3		df-xp 4994	. . . 4 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
4	2, 3	uneq12i 3642	. . 3 |- ( ( A X. C ) u. ( B X. C ) ) = ( { <. x , y >. | ( x e. A /\ y e. C ) } u. { <. x , y >. | ( x e. B /\ y e. C ) } )
5		elun 3631	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 693	. . . . . 6 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A \/ x e. B ) /\ y e. C ) )
7		andir 866	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ y e. C ) <-> ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) )
8	6, 7	bitri 249	. . . . 5 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) )
9	8	opabbii 4503	. . . 4 |- { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) } = { <. x , y >. | ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) }
10		unopab 4514	. . . 4 |- ( { <. x , y >. | ( x e. A /\ y e. C ) } u. { <. x , y >. | ( x e. B /\ y e. C ) } ) = { <. x , y >. | ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) }
11	9, 10	eqtr4i 2486	. . 3 |- { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) } = ( { <. x , y >. | ( x e. A /\ y e. C ) } u. { <. x , y >. | ( x e. B /\ y e. C ) } )
12	4, 11	eqtr4i 2486	. 2 |- ( ( A X. C ) u. ( B X. C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
13	1, 12	eqtr4i 2486	1 |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )

Syntax hints:   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   u. cun 3459   {copab 4496   X. cxp 4986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-opab 4498  df-xp 4994

This theorem is referenced by:  xpun  5046  resundi  5275  xpfi  7783  cdaassen  8553  hashxplem  12478  ustund  20893  cnmpt2pc  21597  pwssplit4  31277  xpprsng  33194