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Theorem ptbasin 22928
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 22923 . . . . 5 (𝑋𝐵 ↔ ∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)))
31elpt 22923 . . . . 5 (𝑌𝐵 ↔ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑏𝑦) ∈ (𝐹𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴𝑑)(𝑏𝑦) = (𝐹𝑦)) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)))
42, 3anbi12i 627 . . . 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 1959 . . . 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 277 . . 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 654 . . . . 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 1445 . . . . . . . . 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 1089 . . . . . . . . 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 274 . . . . . . . 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 3217 . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑎𝑦) = (𝑎𝑘))
13 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑏𝑦) = (𝑏𝑘))
1412, 13ineq12d 4173 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑎𝑦) ∩ (𝑏𝑦)) = ((𝑎𝑘) ∩ (𝑏𝑘)))
1514cbvixpv 8853 . . . . . . . . . . . . . 14 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = X𝑘𝐴 ((𝑎𝑘) ∩ (𝑏𝑘))
16 simpl1l 1224 . . . . . . . . . . . . . . 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 9116 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → (𝑐𝑑) ∈ Fin)
1817ad2antrl 726 . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . . . 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 7035 . . . . . . . . . . . . . . . 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 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 (𝑎𝑦) ∈ (𝐹𝑦))
22 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
2312, 22eleq12d 2832 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑎𝑦) ∈ (𝐹𝑦) ↔ (𝑎𝑘) ∈ (𝐹𝑘)))
2423rspccva 3580 . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑏𝑦) ∈ (𝐹𝑦) ↔ (𝑏𝑘) ∈ (𝐹𝑘)))
2827rspccva 3580 . . . . . . . . . . . . . . . . 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 22248 . . . . . . . . . . . . . . . 16 (((𝐹𝑘) ∈ Top ∧ (𝑎𝑘) ∈ (𝐹𝑘) ∧ (𝑏𝑘) ∈ (𝐹𝑘)) → ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ (𝐹𝑘))
3120, 25, 29, 30syl3anc 1371 . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . . . 20 𝑐 ⊆ (𝑐𝑑)
34 sscon 4098 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐))
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐)
3635sseli 3940 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑐))
3722unieqd 4879 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
3812, 37eqeq12d 2752 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑎𝑦) = (𝐹𝑦) ↔ (𝑎𝑘) = (𝐹𝑘)))
3938rspccva 3580 . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . . . . 20 𝑑 ⊆ (𝑐𝑑)
43 sscon 4098 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑)
4544sseli 3940 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑑))
4613, 37eqeq12d 2752 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑏𝑦) = (𝐹𝑦) ↔ (𝑏𝑘) = (𝐹𝑘)))
4746rspccva 3580 . . . . . . . . . . . . . . . . . 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 4173 . . . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ∩ (𝐹𝑘)) = (𝐹𝑘)
5149, 50eqtrdi 2792 . . . . . . . . . . . . . . 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 22925 . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 3205 . . . . . . . . . . 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 242 . . . . . . . . . 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 1120 . . . . . . . . 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 455 . . . . . . . 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 4167 . . . . . . . . 9 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦)))
61 ixpin 8861 . . . . . . . . 9 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦))
6260, 61eqtr4di 2794 . . . . . . . 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 246 . . . . . 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 454 . . . . 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 241 . . . 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 1937 . . 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 241 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → ((𝑋𝐵𝑌𝐵) → (𝑋𝑌) ∈ 𝐵))
6968imp 407 1 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wral 3064  wrex 3073  cdif 3907  cun 3908  cin 3909  wss 3910   cuni 4865   Fn wfn 6491  wf 6492  cfv 6496  Xcixp 8835  Fincfn 8883  Topctop 22242
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-ixp 8836  df-en 8884  df-fin 8887  df-top 22243
This theorem is referenced by:  ptbasin2  22929
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