Theorem iunin2 2605

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 2598 to recover Enderton's theorem.

Assertion

Ref	Expression
iunin2	|- U_x e. A (B i^i C) = (B i^i U_x e. A C)

Distinct variable group:   x,B

Proof of Theorem iunin2

Step	Hyp	Ref	Expression
1		r19.42v 1763	. . . 4 |- (E.x e. A (y e. B /\ y e. C) <-> (y e. B /\ E.x e. A y e. C))
2		elin 2205	. . . . 5 |- (y e. (B i^i C) <-> (y e. B /\ y e. C))
3	2	rexbii 1667	. . . 4 |- (E.x e. A y e. (B i^i C) <-> E.x e. A (y e. B /\ y e. C))
4		eliun 2567	. . . . 5 |- (y e. U_x e. A C <-> E.x e. A y e. C)
5	4	anbi2i 480	. . . 4 |- ((y e. B /\ y e. U_x e. A C) <-> (y e. B /\ E.x e. A y e. C))
6	1, 3, 5	3bitr4 183	. . 3 |- (E.x e. A y e. (B i^i C) <-> (y e. B /\ y e. U_x e. A C))
7		eliun 2567	. . 3 |- (y e. U_x e. A (B i^i C) <-> E.x e. A y e. (B i^i C))
8		elin 2205	. . 3 |- (y e. (B i^i U_x e. A C) <-> (y e. B /\ y e. U_x e. A C))
9	6, 7, 8	3bitr4 183	. 2 |- (y e. U_x e. A (B i^i C) <-> y e. (B i^i U_x e. A C))
10	9	eqriv 1474	1 |- U_x e. A (B i^i C) = (B i^i U_x e. A C)

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957  E.wrex 1645   i^i cin 2044  U_ciun 2563

This theorem is referenced by:  kmlem11 4762  subtop 7625

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1649  df-v 1810  df-in 2049  df-iun 2565

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