Theorem fiinfi 36897

Description: If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.)

Hypotheses

Ref	Expression
fiinfi.a	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)
fiinfi.b	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)
fiinfi.c	⊢ (𝜑 → 𝐶 = (𝐴 ∩ 𝐵))

Assertion

Ref	Expression
fiinfi	⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Proof of Theorem fiinfi

Step	Hyp	Ref	Expression
1		fiinfi.a	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)
2		elinel1 3761	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴)
3		elinel1 3761	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) → 𝑦 ∈ 𝐴)
4	3	imim1i 61	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐴))
5	4	ralimi2 2933	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴)
6	2, 5	imim12i 60	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴 ∩ 𝐵) → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴))
7	6	ralimi2 2933	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴)
8	1, 7	syl 17	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴)
9		fiinfi.b	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)
10		elinel2 3762	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵)
11		elinel2 3762	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) → 𝑦 ∈ 𝐵)
12	11	imim1i 61	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 → (𝑥 ∩ 𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴 ∩ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵))
13	12	ralimi2 2933	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵)
14	10, 13	imim12i 60	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵))
15	14	ralimi2 2933	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵)
16	9, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵)
17		r19.26-2 3047	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐵))
18	8, 16, 17	sylanbrc 695	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵))
19		elin 3758	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵) ↔ ((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵))
20	19	2ralbii 2964	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)((𝑥 ∩ 𝑦) ∈ 𝐴 ∧ (𝑥 ∩ 𝑦) ∈ 𝐵))
21	18, 20	sylibr 223	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵))
22		fiinfi.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐴 ∩ 𝐵))
23	22	eleq2d 2673	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∩ 𝑦) ∈ 𝐶 ↔ (𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵)))
24	23	ralbidv 2969	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵)))
25	24	ralbidv 2969	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ (𝐴 ∩ 𝐵)))
26	21, 25	mpbird 246	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶)
27	22	raleqdv 3121	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶))
28	27	ralbidv 2969	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥 ∩ 𝑦) ∈ 𝐶))
29	26, 28	mpbird 246	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶)
30	22	raleqdv 3121	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶))
31	29, 30	mpbird 246	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547

This theorem is referenced by: (None)