Theorem eliuniin2 46350

Description: Indexed union of indexed intersections. See eliincex 46340 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
eliuniin2.1	⊢ Ⅎ𝑥𝐶
eliuniin2.2	⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷

Assertion

Ref	Expression
eliuniin2	⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))

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

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

Proof of Theorem eliuniin2

Step	Hyp	Ref	Expression
1		eliuniin2.2	. . . . 5 ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷
2	1	eleq2i 2856	. . . 4 ⊢ (𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
3		eliun 4958	. . . 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 4959	. . . . . 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		eliuniin2.1	. . . 4 ⊢ Ⅎ𝑥𝐶
11		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥∅
12	10, 11	nfne 3062	. . 3 ⊢ Ⅎ𝑥 𝐶 ≠ ∅
13		nfv 1949	. . 3 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐴
14		simp2 1155	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑥 ∈ 𝐵)
15		eliin2 46346	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
16	15	biimpar 483	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
17		rspe 3256	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
18	14, 16, 17	3imp3i2an 1364	. . . . . 6 ⊢ ((𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
19	18, 3	sylibr 237	. . . . 5 ⊢ ((𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
20	19, 2	sylibr 237	. . . 4 ⊢ ((𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐴)
21	20	3exp 1137	. . 3 ⊢ (𝐶 ≠ ∅ → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴)))
22	12, 13, 21	rexlimd 3273	. 2 ⊢ (𝐶 ≠ ∅ → (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴))
23	9, 22	impbid2 229	1 ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∅c0 4281  ∪ ciun 4954  ∩ ciin 4955

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-sbc 3743  df-csb 3852  df-dif 4050  df-nul 4282  df-iun 4956  df-iin 4957

This theorem is used by: (None)