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Theorem fiinfi 38195
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 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
fiinfi.b (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
fiinfi.c (𝜑𝐶 = (𝐴𝐵))
Assertion
Ref Expression
fiinfi (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Proof of Theorem fiinfi
StepHypRef Expression
1 fiinfi.a . . . . . . 7 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2 elinel1 3832 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 elinel1 3832 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐴)
43imim1i 63 . . . . . . . . . 10 ((𝑦𝐴 → (𝑥𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐴))
54ralimi2 2978 . . . . . . . . 9 (∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
62, 5imim12i 62 . . . . . . . 8 ((𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴))
76ralimi2 2978 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
81, 7syl 17 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
9 fiinfi.b . . . . . . 7 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵)
10 elinel2 3833 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐵)
11 elinel2 3833 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐵)
1211imim1i 63 . . . . . . . . . 10 ((𝑦𝐵 → (𝑥𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐵))
1312ralimi2 2978 . . . . . . . . 9 (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
1410, 13imim12i 62 . . . . . . . 8 ((𝑥𝐵 → ∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
1514ralimi2 2978 . . . . . . 7 (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
169, 15syl 17 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
17 r19.26-2 3094 . . . . . 6 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
188, 16, 17sylanbrc 699 . . . . 5 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
19 elin 3829 . . . . . 6 ((𝑥𝑦) ∈ (𝐴𝐵) ↔ ((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
20192ralbii 3010 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
2118, 20sylibr 224 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵))
22 fiinfi.c . . . . . . 7 (𝜑𝐶 = (𝐴𝐵))
2322eleq2d 2716 . . . . . 6 (𝜑 → ((𝑥𝑦) ∈ 𝐶 ↔ (𝑥𝑦) ∈ (𝐴𝐵)))
2423ralbidv 3015 . . . . 5 (𝜑 → (∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2524ralbidv 3015 . . . 4 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2621, 25mpbird 247 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶)
2722raleqdv 3174 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2827ralbidv 3015 . . 3 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2926, 28mpbird 247 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
3022raleqdv 3174 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶))
3129, 30mpbird 247 1 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-in 3614
This theorem is referenced by: (None)
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