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Theorem fiinfi 40119
Description: If two classes have the finite intersection property, then so does their intersection. (Contributed by RP, 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 4155 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 elinel1 4155 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐴)
43imim1i 63 . . . . . . . . . 10 ((𝑦𝐴 → (𝑥𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐴))
54ralimi2 3151 . . . . . . . . 9 (∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
62, 5imim12i 62 . . . . . . . 8 ((𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴))
76ralimi2 3151 . . . . . . 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 4156 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐵)
11 elinel2 4156 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐵)
1211imim1i 63 . . . . . . . . . 10 ((𝑦𝐵 → (𝑥𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐵))
1312ralimi2 3151 . . . . . . . . 9 (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
1410, 13imim12i 62 . . . . . . . 8 ((𝑥𝐵 → ∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
1514ralimi2 3151 . . . . . . 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 3165 . . . . . 6 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
188, 16, 17sylanbrc 586 . . . . 5 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
19 elin 3934 . . . . . 6 ((𝑥𝑦) ∈ (𝐴𝐵) ↔ ((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
20192ralbii 3160 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
2118, 20sylibr 237 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵))
22 fiinfi.c . . . . . . 7 (𝜑𝐶 = (𝐴𝐵))
2322eleq2d 2901 . . . . . 6 (𝜑 → ((𝑥𝑦) ∈ 𝐶 ↔ (𝑥𝑦) ∈ (𝐴𝐵)))
2423ralbidv 3191 . . . . 5 (𝜑 → (∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2524ralbidv 3191 . . . 4 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2621, 25mpbird 260 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶)
2722raleqdv 3402 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2827ralbidv 3191 . . 3 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2926, 28mpbird 260 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
3022raleqdv 3402 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶))
3129, 30mpbird 260 1 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3132  cin 3917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3137  df-v 3481  df-in 3925
This theorem is referenced by: (None)
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