Theorem iunin2 5014

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5002 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 3170	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elin 3906	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3	2	rexbii 3085	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4		eliun 4938	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
5	4	anbi2i 624	. . . 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 4938	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶))
8		elin 3906	. . 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 2734	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3889  ∪ ciun 4934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3432  df-in 3897  df-iun 4936

This theorem is referenced by:  iunin1  5015  2iunin  5019  resiun2  5959  infssuni  9249  kmlem11  10074  cmpsublem  23374  cmpsub  23375  kgentopon  23513  metnrmlem3  24837  ovoliunlem1  25479  voliunlem1  25527  voliunlem2  25528  uniioombllem2  25560  uniioombllem4  25563  volsup2  25582  itg1addlem5  25677  itg1climres  25691  uniin2  32637  carsgclctunlem2  34479  cvmscld  35471  cnambfre  38003  ftc1anclem6  38033  heiborlem3  38148  carageniuncllem2  46968