Theorem iunin2 5029

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5017 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 3195	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elin 3915	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3	2	rexbii 3110	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4		eliun 4955	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
5	4	anbi2i 635	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	1, 3, 5	3bitr4i 306	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
7		eliun 4955	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶))
8		elin 3915	. . 3 ⊢ (𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
9	6, 7, 8	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶))
10	9	eqriv 2758	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∩ cin 3898  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-v 3453  df-in 3906  df-iun 4953

This theorem is used by:  iunin1  5030  uniin2  5034  2iunin  5036  resiun2  5991  infssuni  9319  kmlem11  10220  cmpsublem  23697  cmpsub  23698  kgentopon  23837  metnrmlem3  25161  ovoliunlem1  25803  voliunlem1  25851  voliunlem2  25852  uniioombllem2  25884  uniioombllem4  25887  volsup2  25906  itg1addlem5  26001  itg1climres  26015  carsgclctunlem2  34934  cvmscld  36007  cnambfre  38554  ftc1anclem6  38584  heiborlem3  38715  carageniuncllem2  47476