Theorem 2iunin 3939

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 3936	. . . 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 3891	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
4		iunin1 3937	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
5	3, 4	eqtri 2191	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   = wceq 1348   ∈ wcel 2141   ∩ cin 3120  ∪ ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875

This theorem is referenced by: (None)