Theorem iunin1 3977

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3966 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 3976	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
2		incom 3351	. . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
3	2	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
4	3	iuneq2i 3930	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
5		incom 3351	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
6	1, 4, 5	3eqtr4i 2224	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)

Syntax hints:   = wceq 1364   ∈ wcel 2164   ∩ cin 3152  ∪ ciun 3912

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-iun 3914

This theorem is referenced by:  2iunin  3979  tgrest  14337