Theorem iunin1 5008

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4995 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 5007	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
2		incom 4145	. . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
4	3	iuneq2i 4950	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
5		incom 4145	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
6	1, 4, 5	3eqtr4i 2773	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)

Syntax hints:   = wceq 1547   ∈ wcel 2119   ∩ cin 3889  ∪ ciun 4928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-in 3897  df-ss 3907  df-iun 4930

This theorem is referenced by:  2iunin  5012  resiun1  5958  tgrest  23149  metnrmlem3  24852  limciun  25886  uniin1  32647  disjunsn  32690  measinblem  34411  sstotbnd2  38148  subsaliuncl  46808  sge0iunmptlemre  46865