Theorem 2iunin 4035

Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Assertion

Ref	Expression
2iunin	⊢ ∪x ∈ A ∪y ∈ B (C ∩ D) = (∪x ∈ A C ∩ ∪y ∈ B D)

Distinct variable groups:   x,B   y,C   x,D   x,y

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

Proof of Theorem 2iunin

Step	Hyp	Ref	Expression
1		iunin2 4031	. . . 4 ⊢ ∪y ∈ B (C ∩ D) = (C ∩ ∪y ∈ B D)
2	1	a1i 10	. . 3 ⊢ (x ∈ A → ∪y ∈ B (C ∩ D) = (C ∩ ∪y ∈ B D))
3	2	iuneq2i 3988	. 2 ⊢ ∪x ∈ A ∪y ∈ B (C ∩ D) = ∪x ∈ A (C ∩ ∪y ∈ B D)
4		iunin1 4032	. 2 ⊢ ∪x ∈ A (C ∩ ∪y ∈ B D) = (∪x ∈ A C ∩ ∪y ∈ B D)
5	3, 4	eqtri 2373	1 ⊢ ∪x ∈ A ∪y ∈ B (C ∩ D) = (∪x ∈ A C ∩ ∪y ∈ B D)

Syntax hints:   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  ∪ciun 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 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by: (None)