Theorem 2iunin 4891

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 4886	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
2	1	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷))
3	2	iuneq2i 4839	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
4		iunin1 4887	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
5	3, 4	eqtri 2817	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   = wceq 1520   ∈ wcel 2079   ∩ cin 3853  ∪ ciun 4819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-v 3434  df-in 3861  df-ss 3869  df-iun 4821

This theorem is referenced by:  fpar  7658