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Theorem ptbasin 23601
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 23596 . . . . 5 (𝑋𝐵 ↔ ∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)))
31elpt 23596 . . . . 5 (𝑌𝐵 ↔ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)))
42, 3anbi12i 628 . . . 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 exdistrv 1953 . . . 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 278 . . 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 656 . . . . 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 1444 . . . . . . . . 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 1088 . . . . . . . . 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 275 . . . . . . . 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 3227 . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑎𝑦) = (𝑎𝑘))
13 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑏𝑦) = (𝑏𝑘))
1412, 13ineq12d 4229 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑎𝑦) ∩ (𝑏𝑦)) = ((𝑎𝑘) ∩ (𝑏𝑘)))
1514cbvixpv 8954 . . . . . . . . . . . . . 14 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = X𝑘𝐴 ((𝑎𝑘) ∩ (𝑏𝑘))
16 simpl1l 1223 . . . . . . . . . . . . . . 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 9210 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → (𝑐𝑑) ∈ Fin)
1817ad2antrl 728 . . . . . . . . . . . . . . 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 1224 . . . . . . . . . . . . . . . . 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)
2019ffvelcdmda 7104 . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
2312, 22eleq12d 2833 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑎𝑦) ∈ (𝐹𝑦) ↔ (𝑎𝑘) ∈ (𝐹𝑘)))
2423rspccva 3621 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ 𝑘𝐴) → (𝑎𝑘) ∈ (𝐹𝑘))
2521, 24sylan 580 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑏𝑦) ∈ (𝐹𝑦) ↔ (𝑏𝑘) ∈ (𝐹𝑘)))
2827rspccva 3621 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ 𝑘𝐴) → (𝑏𝑘) ∈ (𝐹𝑘))
2926, 28sylan 580 . . . . . . . . . . . . . . . 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 22921 . . . . . . . . . . . . . . . 16 (((𝐹𝑘) ∈ Top ∧ (𝑎𝑘) ∈ (𝐹𝑘) ∧ (𝑏𝑘) ∈ (𝐹𝑘)) → ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ (𝐹𝑘))
3120, 25, 29, 30syl3anc 1370 . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . 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 4188 . . . . . . . . . . . . . . . . . . . 20 𝑐 ⊆ (𝑐𝑑)
34 sscon 4153 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐))
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐)
3635sseli 3991 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑐))
3722unieqd 4925 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
3812, 37eqeq12d 2751 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑎𝑦) = (𝐹𝑦) ↔ (𝑎𝑘) = (𝐹𝑘)))
3938rspccva 3621 . . . . . . . . . . . . . . . . . 18 ((∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦) ∧ 𝑘 ∈ (𝐴𝑐)) → (𝑎𝑘) = (𝐹𝑘))
4032, 36, 39syl2an 596 . . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . 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 4189 . . . . . . . . . . . . . . . . . . . 20 𝑑 ⊆ (𝑐𝑑)
43 sscon 4153 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑)
4544sseli 3991 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑑))
4613, 37eqeq12d 2751 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑏𝑦) = (𝐹𝑦) ↔ (𝑏𝑘) = (𝐹𝑘)))
4746rspccva 3621 . . . . . . . . . . . . . . . . . 18 ((∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦) ∧ 𝑘 ∈ (𝐴𝑑)) → (𝑏𝑘) = (𝐹𝑘))
4841, 45, 47syl2an 596 . . . . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . 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 4235 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ∩ (𝐹𝑘)) = (𝐹𝑘)
5149, 50eqtrdi 2791 . . . . . . . . . . . . . . 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 23598 . . . . . . . . . . . . . 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, 52eqeltrid 2843 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 3211 . . . . . . . . . . 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, 55biimtrrid 243 . . . . . . . . . 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 1119 . . . . . . . . 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 454 . . . . . . . 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 594 . . . . . . 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 4223 . . . . . . . . 9 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦)))
61 ixpin 8962 . . . . . . . . 9 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦))
6260, 61eqtr4di 2793 . . . . . . . 8 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)))
6362eleq1d 2824 . . . . . . 7 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → ((𝑋𝑌) ∈ 𝐵X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) ∈ 𝐵))
6459, 63syl5ibrcom 247 . . . . . 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 453 . . . . 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, 65biimtrid 242 . . . 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 1932 . . 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, 67biimtrid 242 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → ((𝑋𝐵𝑌𝐵) → (𝑋𝑌) ∈ 𝐵))
6968imp 406 1 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  cdif 3960  cun 3961  cin 3962  wss 3963   cuni 4912   Fn wfn 6558  wf 6559  cfv 6563  Xcixp 8936  Fincfn 8984  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-ixp 8937  df-en 8985  df-fin 8988  df-top 22916
This theorem is referenced by:  ptbasin2  23602
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