Theorem 2iunin 4042

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 4039	. . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
2	1	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷))
3	2	iuneq2i 3993	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
4		iunin1 4040	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
5	3, 4	eqtri 2252	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   = wceq 1398   ∈ wcel 2202   ∩ cin 3200  ∪ ciun 3975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-iun 3977

This theorem is referenced by: (None)