Theorem coundir 5084

Description: Class composition distributes over union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
coundir	|- ((A u. B) o. C) = ((A o. C) u. (B o. C))

Proof of Theorem coundir

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

Step	Hyp	Ref	Expression
1		unopab 4639	. . 3 |- ({<.x, z>. | E.y(xCy /\ yAz)} u. {<.x, z>. | E.y(xCy /\ yBz)}) = {<.x, z>. | (E.y(xCy /\ yAz) \/ E.y(xCy /\ yBz))}
2		brun 4693	. . . . . . . 8 |- (y(A u. B)z <-> (yAz \/ yBz))
3	2	anbi2i 675	. . . . . . 7 |- ((xCy /\ y(A u. B)z) <-> (xCy /\ (yAz \/ yBz)))
4		andi 837	. . . . . . 7 |- ((xCy /\ (yAz \/ yBz)) <-> ((xCy /\ yAz) \/ (xCy /\ yBz)))
5	3, 4	bitri 240	. . . . . 6 |- ((xCy /\ y(A u. B)z) <-> ((xCy /\ yAz) \/ (xCy /\ yBz)))
6	5	exbii 1582	. . . . 5 |- (E.y(xCy /\ y(A u. B)z) <-> E.y((xCy /\ yAz) \/ (xCy /\ yBz)))
7		19.43 1605	. . . . 5 |- (E.y((xCy /\ yAz) \/ (xCy /\ yBz)) <-> (E.y(xCy /\ yAz) \/ E.y(xCy /\ yBz)))
8	6, 7	bitr2i 241	. . . 4 |- ((E.y(xCy /\ yAz) \/ E.y(xCy /\ yBz)) <-> E.y(xCy /\ y(A u. B)z))
9	8	opabbii 4627	. . 3 |- {<.x, z>. | (E.y(xCy /\ yAz) \/ E.y(xCy /\ yBz))} = {<.x, z>. | E.y(xCy /\ y(A u. B)z)}
10	1, 9	eqtri 2373	. 2 |- ({<.x, z>. | E.y(xCy /\ yAz)} u. {<.x, z>. | E.y(xCy /\ yBz)}) = {<.x, z>. | E.y(xCy /\ y(A u. B)z)}
11		df-co 4727	. . 3 |- (A o. C) = {<.x, z>. | E.y(xCy /\ yAz)}
12		df-co 4727	. . 3 |- (B o. C) = {<.x, z>. | E.y(xCy /\ yBz)}
13	11, 12	uneq12i 3417	. 2 |- ((A o. C) u. (B o. C)) = ({<.x, z>. | E.y(xCy /\ yAz)} u. {<.x, z>. | E.y(xCy /\ yBz)})
14		df-co 4727	. 2 |- ((A u. B) o. C) = {<.x, z>. | E.y(xCy /\ y(A u. B)z)}
15	10, 13, 14	3eqtr4ri 2384	1 |- ((A u. B) o. C) = ((A o. C) u. (B o. C))

Syntax hints:   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   u. cun 3208  {copab 4623   class class class wbr 4640   o. ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by: (None)