Theorem eliuniin 45676

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 2857	. . . 4 ⊢ (𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
3		eliun 4955	. . . 4 ⊢ (𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
4	2, 3	sylbb 222	. . 3 ⊢ (𝑍 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
5		eliin 4956	. . . . . 6 ⊢ (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
6	5	ibi 270	. . . . 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 3180	. . 3 ⊢ (𝑍 ∈ 𝐴 → (∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
9	4, 8	mpd 16	. 2 ⊢ (𝑍 ∈ 𝐴 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
10		simp2 1153	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑥 ∈ 𝐵)
11		eliin 4956	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
12	11	biimpar 482	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
13		rspe 3255	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
14	10, 12, 13	3imp3i2an 1362	. . . . 5 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
15	14, 3	sylibr 237	. . . 4 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
16	15, 2	sylibr 237	. . 3 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐴)
17	16	rexlimdv3a 3170	. 2 ⊢ (𝑍 ∈ 𝑉 → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴))
18	9, 17	impbid2 229	1 ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∪ ciun 4951  ∩ ciin 4952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-v 3459  df-iun 4953  df-iin 4954

This theorem is referenced by: (None)