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Theorem ptbasin 21580
 Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
Assertion
Ref Expression
ptbasin (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)
Distinct variable groups:   𝑥,𝑔,𝑦,𝑧,𝐴   𝑔,𝑌,𝑥   𝑔,𝐹,𝑥,𝑦,𝑧   𝑔,𝑋,𝑥,𝑧   𝑔,𝑉,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑔)   𝑋(𝑦)   𝑌(𝑦,𝑧)

Proof of Theorem ptbasin
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptbas.1 . . . . . 6 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
21elpt 21575 . . . . 5 (𝑋𝐵 ↔ ∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)))
31elpt 21575 . . . . 5 (𝑌𝐵 ↔ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)))
42, 3anbi12i 735 . . . 4 ((𝑋𝐵𝑌𝐵) ↔ (∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))))
5 eeanv 2325 . . . 4 (∃𝑎𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))) ↔ (∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))))
64, 5bitr4i 267 . . 3 ((𝑋𝐵𝑌𝐵) ↔ ∃𝑎𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))))
7 an4 900 . . . . 5 ((((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))) ↔ (((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))) ∧ (𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))))
8 an6 1555 . . . . . . . . 9 (((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))) ↔ ((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦)) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))))
9 df-3an 1074 . . . . . . . . 9 (((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦)) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))) ↔ (((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))))
108, 9bitri 264 . . . . . . . 8 (((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))) ↔ (((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))))
11 reeanv 3243 . . . . . . . . . . 11 (∃𝑐 ∈ Fin ∃𝑑 ∈ Fin (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ↔ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))
12 fveq2 6350 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑎𝑦) = (𝑎𝑘))
13 fveq2 6350 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑏𝑦) = (𝑏𝑘))
1412, 13ineq12d 3956 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑎𝑦) ∩ (𝑏𝑦)) = ((𝑎𝑘) ∩ (𝑏𝑘)))
1514cbvixpv 8090 . . . . . . . . . . . . . 14 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = X𝑘𝐴 ((𝑎𝑘) ∩ (𝑏𝑘))
16 simpl1l 1279 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → 𝐴𝑉)
17 unfi 8390 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → (𝑐𝑑) ∈ Fin)
1817ad2antrl 766 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → (𝑐𝑑) ∈ Fin)
19 simpl1r 1281 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → 𝐹:𝐴⟶Top)
2019ffvelrnda 6520 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ Top)
21 simpl3l 1287 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦))
22 fveq2 6350 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
2312, 22eleq12d 2831 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑎𝑦) ∈ (𝐹𝑦) ↔ (𝑎𝑘) ∈ (𝐹𝑘)))
2423rspccva 3446 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ 𝑘𝐴) → (𝑎𝑘) ∈ (𝐹𝑘))
2521, 24sylan 489 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘𝐴) → (𝑎𝑘) ∈ (𝐹𝑘))
26 simpl3r 1289 . . . . . . . . . . . . . . . . 17 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))
2713, 22eleq12d 2831 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑏𝑦) ∈ (𝐹𝑦) ↔ (𝑏𝑘) ∈ (𝐹𝑘)))
2827rspccva 3446 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ 𝑘𝐴) → (𝑏𝑘) ∈ (𝐹𝑘))
2926, 28sylan 489 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘𝐴) → (𝑏𝑘) ∈ (𝐹𝑘))
30 inopn 20904 . . . . . . . . . . . . . . . 16 (((𝐹𝑘) ∈ Top ∧ (𝑎𝑘) ∈ (𝐹𝑘) ∧ (𝑏𝑘) ∈ (𝐹𝑘)) → ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ (𝐹𝑘))
3120, 25, 29, 30syl3anc 1477 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘𝐴) → ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ (𝐹𝑘))
32 simprrl 823 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦))
33 ssun1 3917 . . . . . . . . . . . . . . . . . . . 20 𝑐 ⊆ (𝑐𝑑)
34 sscon 3885 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐))
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐)
3635sseli 3738 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑐))
3722unieqd 4596 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
3812, 37eqeq12d 2773 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑎𝑦) = (𝐹𝑦) ↔ (𝑎𝑘) = (𝐹𝑘)))
3938rspccva 3446 . . . . . . . . . . . . . . . . . 18 ((∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ 𝑘 ∈ (𝐴𝑐)) → (𝑎𝑘) = (𝐹𝑘))
4032, 36, 39syl2an 495 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐𝑑))) → (𝑎𝑘) = (𝐹𝑘))
41 simprrr 824 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))
42 ssun2 3918 . . . . . . . . . . . . . . . . . . . 20 𝑑 ⊆ (𝑐𝑑)
43 sscon 3885 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑)
4544sseli 3738 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑑))
4613, 37eqeq12d 2773 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑏𝑦) = (𝐹𝑦) ↔ (𝑏𝑘) = (𝐹𝑘)))
4746rspccva 3446 . . . . . . . . . . . . . . . . . 18 ((∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦) ∧ 𝑘 ∈ (𝐴𝑑)) → (𝑏𝑘) = (𝐹𝑘))
4841, 45, 47syl2an 495 . . . . . . . . . . . . . . . . 17 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐𝑑))) → (𝑏𝑘) = (𝐹𝑘))
4940, 48ineq12d 3956 . . . . . . . . . . . . . . . 16 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐𝑑))) → ((𝑎𝑘) ∩ (𝑏𝑘)) = ( (𝐹𝑘) ∩ (𝐹𝑘)))
50 inidm 3963 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ∩ (𝐹𝑘)) = (𝐹𝑘)
5149, 50syl6eq 2808 . . . . . . . . . . . . . . 15 (((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐𝑑))) → ((𝑎𝑘) ∩ (𝑏𝑘)) = (𝐹𝑘))
521, 16, 18, 31, 51elptr2 21577 . . . . . . . . . . . . . 14 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → X𝑘𝐴 ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ 𝐵)
5315, 52syl5eqel 2841 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵)
5453expr 644 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin)) → ((∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
5554rexlimdvva 3174 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) → (∃𝑐 ∈ Fin ∃𝑑 ∈ Fin (∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
5611, 55syl5bir 233 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) → ((∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
57563expb 1114 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦)))) → ((∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
5857impr 650 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (((𝑎 Fn 𝐴𝑏 Fn 𝐴) ∧ (∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵)
5910, 58sylan2b 493 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵)
60 ineq12 3950 . . . . . . . . 9 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦)))
61 ixpin 8097 . . . . . . . . 9 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦))
6260, 61syl6eqr 2810 . . . . . . . 8 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)))
6362eleq1d 2822 . . . . . . 7 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → ((𝑋𝑌) ∈ 𝐵X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
6459, 63syl5ibrcom 237 . . . . . 6 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)))) → ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) ∈ 𝐵))
6564expimpd 630 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ((((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦))) ∧ (𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))) → (𝑋𝑌) ∈ 𝐵))
667, 65syl5bi 232 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → ((((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))) → (𝑋𝑌) ∈ 𝐵))
6766exlimdvv 2009 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (∃𝑎𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦))) → (𝑋𝑌) ∈ 𝐵))
686, 67syl5bi 232 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → ((𝑋𝐵𝑌𝐵) → (𝑋𝑌) ∈ 𝐵))
6968imp 444 1 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1630  ∃wex 1851   ∈ wcel 2137  {cab 2744  ∀wral 3048  ∃wrex 3049   ∖ cdif 3710   ∪ cun 3711   ∩ cin 3712   ⊆ wss 3713  ∪ cuni 4586   Fn wfn 6042  ⟶wf 6043  ‘cfv 6047  Xcixp 8072  Fincfn 8119  Topctop 20898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-oadd 7731  df-er 7909  df-ixp 8073  df-en 8120  df-fin 8123  df-top 20899 This theorem is referenced by:  ptbasin2  21581
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