Theorem eliuniin 46387

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 2856	. . . 4 ⊢ (𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
3		eliun 4960	. . . 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 4961	. . . . . 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 3181	. . 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 1155	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑥 ∈ 𝐵)
11		eliin 4961	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
12	11	biimpar 483	. . . . . 6 ⊢ ((𝑍 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
13		rspe 3256	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
14	10, 12, 13	3imp3i2an 1364	. . . . 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 3171	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∪ ciun 4956  ∩ ciin 4957

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-iun 4958  df-iin 4959

This theorem is used by: (None)