Theorem iuniin 3979

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	|- U_x e. A |^|_y e. B C C_ |^|_y e. B U_x e. A C

Distinct variable groups:   x,y   y,A   x,B

Allowed substitution hints:   A(x)   B(y)   C(x,y)

Proof of Theorem iuniin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 2727	. . . 4 |- (E.x e. A A.y e. B z e. C -> A.y e. B E.x e. A z e. C)
2		vex 2862	. . . . . 6 |- z e. _V
3		eliin 3974	. . . . . 6 |- (z e. _V -> (z e. |^|_y e. B C <-> A.y e. B z e. C))
4	2, 3	ax-mp 8	. . . . 5 |- (z e. |^|_y e. B C <-> A.y e. B z e. C)
5	4	rexbii 2639	. . . 4 |- (E.x e. A z e. |^|_y e. B C <-> E.x e. A A.y e. B z e. C)
6		eliun 3973	. . . . 5 |- (z e. U_x e. A C <-> E.x e. A z e. C)
7	6	ralbii 2638	. . . 4 |- (A.y e. B z e. U_x e. A C <-> A.y e. B E.x e. A z e. C)
8	1, 5, 7	3imtr4i 257	. . 3 |- (E.x e. A z e. |^|_y e. B C -> A.y e. B z e. U_x e. A C)
9		eliun 3973	. . 3 |- (z e. U_x e. A |^|_y e. B C <-> E.x e. A z e. |^|_y e. B C)
10		eliin 3974	. . . 4 |- (z e. _V -> (z e. |^|_y e. B U_x e. A C <-> A.y e. B z e. U_x e. A C))
11	2, 10	ax-mp 8	. . 3 |- (z e. |^|_y e. B U_x e. A C <-> A.y e. B z e. U_x e. A C)
12	8, 9, 11	3imtr4i 257	. 2 |- (z e. U_x e. A |^|_y e. B C -> z e. |^|_y e. B U_x e. A C)
13	12	ssriv 3277	1 |- U_x e. A |^|_y e. B C C_ |^|_y e. B U_x e. A C

Syntax hints:   <-> wb 176   e. wcel 1710  A.wral 2614  E.wrex 2615  _Vcvv 2859   C_ wss 3257  U_ciun 3969  |^|_ciin 3970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971  df-iin 3972

This theorem is referenced by: (None)