Theorem iunin1 3930

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3919 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1	⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 3929	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
2		incom 3314	. . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
3	2	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
4	3	iuneq2i 3884	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
5		incom 3314	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
6	1, 4, 5	3eqtr4i 2196	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)

Syntax hints:   = wceq 1343   ∈ wcel 2136   ∩ cin 3115  ∪ ciun 3866

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868

This theorem is referenced by:  2iunin  3932  tgrest  12809