Theorem eliuniin2 45696

Description: Indexed union of indexed intersections. See eliincex 45686 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 2857	. . . 4 ⊢ (𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷)
3		eliun 4956	. . . 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 4957	. . . . . 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		eliuniin2.1	. . . 4 ⊢ Ⅎ𝑥𝐶
11		nfcv 2927	. . . 4 ⊢ Ⅎ𝑥∅
12	10, 11	nfne 3061	. . 3 ⊢ Ⅎ𝑥 𝐶 ≠ ∅
13		nfv 1937	. . 3 ⊢ Ⅎ𝑥 𝑍 ∈ 𝐴
14		simp2 1153	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑥 ∈ 𝐵)
15		eliin2 45692	. . . . . . . 8 ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
16	15	biimpar 482	. . . . . . 7 ⊢ ((𝐶 ≠ ∅ ∧ ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
17		rspe 3255	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷) → ∃𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷)
18	14, 16, 17	3imp3i2an 1362	. . . . . 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 1135	. . 3 ⊢ (𝐶 ≠ ∅ → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴)))
22	12, 13, 21	rexlimd 3272	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Ⅎwnfc 2912   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  ∅c0 4288  ∪ ciun 4952  ∩ ciin 4953

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-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-nul 4289  df-iun 4954  df-iin 4955

This theorem is referenced by: (None)