Theorem iunin1 4031

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4019 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 4030	. 2 ⊢ ∪x ∈ A (B ∩ C) = (B ∩ ∪x ∈ A C)
2		incom 3448	. . . 4 ⊢ (C ∩ B) = (B ∩ C)
3	2	a1i 10	. . 3 ⊢ (x ∈ A → (C ∩ B) = (B ∩ C))
4	3	iuneq2i 3987	. 2 ⊢ ∪x ∈ A (C ∩ B) = ∪x ∈ A (B ∩ C)
5		incom 3448	. 2 ⊢ (∪x ∈ A C ∩ B) = (B ∩ ∪x ∈ A C)
6	1, 4, 5	3eqtr4i 2383	1 ⊢ ∪x ∈ A (C ∩ B) = (∪x ∈ A C ∩ B)

Syntax hints:   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∪ciun 3969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971

This theorem is referenced by:  2iunin  4034