Theorem intun 2566

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.

Assertion

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

Proof of Theorem intun

Step	Hyp	Ref	Expression
1		19.26 1069	. . . 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 2176	. . . . . . 7 |- (y e. (A u. B) <-> (y e. A \/ y e. B))
3	2	imbi1i 186	. . . . . 6 |- ((y e. (A u. B) -> x e. y) <-> ((y e. A \/ y e. B) -> x e. y))
4		jaob 424	. . . . . 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	bitr 173	. . . . 5 |- ((y e. (A u. B) -> x e. y) <-> ((y e. A -> x e. y) /\ (y e. B -> x e. y)))
6	5	albii 1001	. . . 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		visset 1816	. . . . . 6 |- x e. V
8	7	elint 2543	. . . . 5 |- (x e. |^|A <-> A.y(y e. A -> x e. y))
9	7	elint 2543	. . . . 5 |- (x e. |^|B <-> A.y(y e. B -> x e. y))
10	8, 9	anbi12i 484	. . . 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	3bitr4 183	. . 3 |- (A.y(y e. (A u. B) -> x e. y) <-> (x e. |^|A /\ x e. |^|B))
12	7	elint 2543	. . 3 |- (x e. |^|(A u. B) <-> A.y(y e. (A u. B) -> x e. y))
13		elin 2210	. . 3 |- (x e. (|^|A i^i |^|B) <-> (x e. |^|A /\ x e. |^|B))
14	11, 12, 13	3bitr4 183	. 2 |- (x e. |^|(A u. B) <-> x e. (|^|A i^i |^|B))
15	14	eqriv 1477	1 |- |^|(A u. B) = (|^|A i^i |^|B)

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960   u. cun 2048   i^i cin 2049  |^|cint 2537

This theorem is referenced by:  intunsn 2569  infi1 10441  moec 10451  ficli 10462  infi 10559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-int 2538

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