Theorem eliuniin 45553

Description: Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
eliuniin.1	⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷

Assertion

Ref	Expression
eliuniin	⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝑥,𝑍   𝑦,𝑍

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem eliuniin

Step	Hyp	Ref	Expression
1		eliuniin.1	. . . . 5 ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷
2	1	eleq2i 2832	. . . 4 ⊢ (𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
3		eliun 4932	. . . 4 ⊢ (𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
4	2, 3	sylbb 220	. . 3 ⊢ (𝑍 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
5		eliin 4933	. . . . . 6 ⊢ (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
6	5	ibi 268	. . . . 5 ⊢ (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
7	6	a1i 11	. . . 4 ⊢ (𝑍 ∈ 𝐴 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
8	7	reximdv 3155	. . 3 ⊢ (𝑍 ∈ 𝐴 → (∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
9	4, 8	mpd 15	. 2 ⊢ (𝑍 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
10		simp2 1143	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑥 ∈ 𝐵)
11		eliin 4933	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
12	11	biimpar 478	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
13		rspe 3230	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
14	10, 12, 13	3imp3i2an 1352	. . . . 5 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
15	14, 3	sylibr 235	. . . 4 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
16	15, 2	sylibr 235	. . 3 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐴)
17	16	rexlimdv3a 3145	. 2 ⊢ (𝑍 ∈ 𝑉 → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴))
18	9, 17	impbid2 227	1 ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ∪ ciun 4928  ∩ ciin 4929

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-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  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-v 3434  df-iun 4930  df-iin 4931

This theorem is referenced by: (None)