Theorem iinuni 4049

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iinuni	|- (A u. |^|B) = |^|_x e. B (A u. x)

Distinct variable groups:   x,A   x,B

Proof of Theorem iinuni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 2757	. . . 4 |- (A.x e. B (y e. A \/ y e. x) <-> (y e. A \/ A.x e. B y e. x))
2		elun 3220	. . . . 5 |- (y e. (A u. x) <-> (y e. A \/ y e. x))
3	2	ralbii 2638	. . . 4 |- (A.x e. B y e. (A u. x) <-> A.x e. B (y e. A \/ y e. x))
4		vex 2862	. . . . . 6 |- y e. _V
5	4	elint2 3933	. . . . 5 |- (y e. |^|B <-> A.x e. B y e. x)
6	5	orbi2i 505	. . . 4 |- ((y e. A \/ y e. |^|B) <-> (y e. A \/ A.x e. B y e. x))
7	1, 3, 6	3bitr4ri 269	. . 3 |- ((y e. A \/ y e. |^|B) <-> A.x e. B y e. (A u. x))
8		elun 3220	. . 3 |- (y e. (A u. |^|B) <-> (y e. A \/ y e. |^|B))
9		eliin 3974	. . . 4 |- (y e. _V -> (y e. |^|_x e. B (A u. x) <-> A.x e. B y e. (A u. x)))
10	4, 9	ax-mp 5	. . 3 |- (y e. |^|_x e. B (A u. x) <-> A.x e. B y e. (A u. x))
11	7, 8, 10	3bitr4i 268	. 2 |- (y e. (A u. |^|B) <-> y e. |^|_x e. B (A u. x))
12	11	eqriv 2350	1 |- (A u. |^|B) = |^|_x e. B (A u. x)

Syntax hints:   <-> wb 176   \/ wo 357   = wceq 1642   e. wcel 1710  A.wral 2614  _Vcvv 2859   u. cun 3207  |^|cint 3926  |^|_ciin 3970

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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-int 3927  df-iin 3972

This theorem is referenced by: (None)