Theorem intun 3959

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Assertion

Ref	Expression
intun	|- |^|(A u. B) = (|^|A i^i |^|B)

Proof of Theorem intun

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

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 |- (A.y((y e. A -> x e. y) /\ (y e. B -> x e. y)) <-> (A.y(y e. A -> x e. y) /\ A.y(y e. B -> x e. y)))
2		elun 3221	. . . . . . 7 |- (y e. (A u. B) <-> (y e. A \/ y e. B))
3	2	imbi1i 315	. . . . . 6 |- ((y e. (A u. B) -> x e. y) <-> ((y e. A \/ y e. B) -> x e. y))
4		jaob 758	. . . . . 6 |- (((y e. A \/ y e. B) -> x e. y) <-> ((y e. A -> x e. y) /\ (y e. B -> x e. y)))
5	3, 4	bitri 240	. . . . 5 |- ((y e. (A u. B) -> x e. y) <-> ((y e. A -> x e. y) /\ (y e. B -> x e. y)))
6	5	albii 1566	. . . 4 |- (A.y(y e. (A u. B) -> x e. y) <-> A.y((y e. A -> x e. y) /\ (y e. B -> x e. y)))
7		vex 2863	. . . . . 6 |- x e. _V
8	7	elint 3933	. . . . 5 |- (x e. |^|A <-> A.y(y e. A -> x e. y))
9	7	elint 3933	. . . . 5 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
10	8, 9	anbi12i 678	. . . 4 |- ((x e. |^|A /\ x e. |^|B) <-> (A.y(y e. A -> x e. y) /\ A.y(y e. B -> x e. y)))
11	1, 6, 10	3bitr4i 268	. . 3 |- (A.y(y e. (A u. B) -> x e. y) <-> (x e. |^|A /\ x e. |^|B))
12	7	elint 3933	. . 3 |- (x e. |^|(A u. B) <-> A.y(y e. (A u. B) -> x e. y))
13		elin 3220	. . 3 |- (x e. (|^|A i^i |^|B) <-> (x e. |^|A /\ x e. |^|B))
14	11, 12, 13	3bitr4i 268	. 2 |- (x e. |^|(A u. B) <-> x e. (|^|A i^i |^|B))
15	14	eqriv 2350	1 |- |^|(A u. B) = (|^|A i^i |^|B)

Syntax hints:   -> wi 4   \/ wo 357   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   u. cun 3208   i^i cin 3209  |^|cint 3927

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-in 3214  df-un 3215  df-int 3928

This theorem is referenced by:  intunsn  3966