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Theorem fiinfi 36693
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 3760 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
3 elinel1 3760 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐴)
43imim1i 60 . . . . . . . . . 10 ((𝑦𝐴 → (𝑥𝑦) ∈ 𝐴) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐴))
54ralimi2 2932 . . . . . . . . 9 (∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴)
62, 5imim12i 59 . . . . . . . 8 ((𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴))
76ralimi2 2932 . . . . . . 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 3761 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐵)
11 elinel2 3761 . . . . . . . . . . 11 (𝑦 ∈ (𝐴𝐵) → 𝑦𝐵)
1211imim1i 60 . . . . . . . . . 10 ((𝑦𝐵 → (𝑥𝑦) ∈ 𝐵) → (𝑦 ∈ (𝐴𝐵) → (𝑥𝑦) ∈ 𝐵))
1312ralimi2 2932 . . . . . . . . 9 (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵)
1410, 13imim12i 59 . . . . . . . 8 ((𝑥𝐵 → ∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → (𝑥 ∈ (𝐴𝐵) → ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
1514ralimi2 2932 . . . . . . 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 3046 . . . . . 6 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵) ↔ (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐴 ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐵))
188, 16, 17sylanbrc 694 . . . . 5 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
19 elin 3757 . . . . . 6 ((𝑥𝑦) ∈ (𝐴𝐵) ↔ ((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
20192ralbii 2963 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵) ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)((𝑥𝑦) ∈ 𝐴 ∧ (𝑥𝑦) ∈ 𝐵))
2118, 20sylibr 222 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵))
22 fiinfi.c . . . . . . 7 (𝜑𝐶 = (𝐴𝐵))
2322eleq2d 2672 . . . . . 6 (𝜑 → ((𝑥𝑦) ∈ 𝐶 ↔ (𝑥𝑦) ∈ (𝐴𝐵)))
2423ralbidv 2968 . . . . 5 (𝜑 → (∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2524ralbidv 2968 . . . 4 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ (𝐴𝐵)))
2621, 25mpbird 245 . . 3 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶)
2722raleqdv 3120 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2827ralbidv 2968 . . 3 (𝜑 → (∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥𝑦) ∈ 𝐶))
2926, 28mpbird 245 . 2 (𝜑 → ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
3022raleqdv 3120 . 2 (𝜑 → (∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐴𝐵)∀𝑦𝐶 (𝑥𝑦) ∈ 𝐶))
3129, 30mpbird 245 1 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥𝑦) ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  cin 3538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-in 3546
This theorem is referenced by: (None)
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