Theorem 2iunin 3796

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 3793	. . . 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 3748	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
4		iunin1 3794	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
5	3, 4	eqtri 2108	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   = wceq 1289   ∈ wcel 1438   ∩ cin 2998  ∪ ciun 3730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-iun 3732

This theorem is referenced by: (None)