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Theorem fiinfi 38556
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 3963 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 elinel1 3963 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐴)
43imim1i 63 . . . . . . . . . 10 ((𝑦𝐴 → (𝑥𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐴))
54ralimi2 3096 . . . . . . . . 9 (∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
62, 5imim12i 62 . . . . . . . 8 ((𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴))
76ralimi2 3096 . . . . . . 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 3964 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐵)
11 elinel2 3964 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐵)
1211imim1i 63 . . . . . . . . . 10 ((𝑦𝐵 → (𝑥𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐵))
1312ralimi2 3096 . . . . . . . . 9 (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
1410, 13imim12i 62 . . . . . . . 8 ((𝑥𝐵 → ∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
1514ralimi2 3096 . . . . . . 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 3212 . . . . . 6 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
188, 16, 17sylanbrc 578 . . . . 5 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
19 elin 3960 . . . . . 6 ((𝑥𝑦) ∈ (𝐴𝐵) ↔ ((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
20192ralbii 3128 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
2118, 20sylibr 225 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵))
22 fiinfi.c . . . . . . 7 (𝜑𝐶 = (𝐴𝐵))
2322eleq2d 2830 . . . . . 6 (𝜑 → ((𝑥𝑦) ∈ 𝐶 ↔ (𝑥𝑦) ∈ (𝐴𝐵)))
2423ralbidv 3133 . . . . 5 (𝜑 → (∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2524ralbidv 3133 . . . 4 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2621, 25mpbird 248 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶)
2722raleqdv 3292 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2827ralbidv 3133 . . 3 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2926, 28mpbird 248 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
3022raleqdv 3292 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶))
3129, 30mpbird 248 1 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  cin 3733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-in 3741
This theorem is referenced by: (None)
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