Theorem iunin2 5026

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5014 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
iunin2	⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝑥,𝐵

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

Proof of Theorem iunin2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.42v 3168	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elin 3917	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3	2	rexbii 3083	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4		eliun 4950	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
5	4	anbi2i 623	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	1, 3, 5	3bitr4i 303	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
7		eliun 4950	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶))
8		elin 3917	. . 3 ⊢ (𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	6, 7, 8	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶))
10	9	eqriv 2733	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ∩ cin 3900  ∪ ciun 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-v 3442  df-in 3908  df-iun 4948

This theorem is referenced by:  iunin1  5027  2iunin  5031  resiun2  5959  infssuni  9246  kmlem11  10071  cmpsublem  23343  cmpsub  23344  kgentopon  23482  metnrmlem3  24806  ovoliunlem1  25459  voliunlem1  25507  voliunlem2  25508  uniioombllem2  25540  uniioombllem4  25543  volsup2  25562  itg1addlem5  25657  itg1climres  25671  uniin2  32627  carsgclctunlem2  34476  cvmscld  35467  cnambfre  37865  ftc1anclem6  37895  heiborlem3  38010  carageniuncllem2  46762