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Theorem ptbasin 21503
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 21498 . . . . 5 (𝑋𝐵 ↔ ∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑎𝑦) ∈ (𝐹𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴𝑐)(𝑎𝑦) = (𝐹𝑦)) ∧ 𝑋 = X𝑦𝐴 (𝑎𝑦)))
31elpt 21498 . . . . 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 2291 . . . 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 1521 . . . . . . . . 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 3209 . . . . . . . . . . 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 6304 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑎𝑦) = (𝑎𝑘))
13 fveq2 6304 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑘 → (𝑏𝑦) = (𝑏𝑘))
1412, 13ineq12d 3923 . . . . . . . . . . . . . . 15 (𝑦 = 𝑘 → ((𝑎𝑦) ∩ (𝑏𝑦)) = ((𝑎𝑘) ∩ (𝑏𝑘)))
1514cbvixpv 8043 . . . . . . . . . . . . . 14 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = X𝑘𝐴 ((𝑎𝑘) ∩ (𝑏𝑘))
16 simpl1l 1255 . . . . . . . . . . . . . . 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 8343 . . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . . 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 6474 . . . . . . . . . . . . . . . 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 1263 . . . . . . . . . . . . . . . . 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 6304 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
2312, 22eleq12d 2797 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑎𝑦) ∈ (𝐹𝑦) ↔ (𝑎𝑘) ∈ (𝐹𝑘)))
2423rspccva 3412 . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . . 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 2797 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑘 → ((𝑏𝑦) ∈ (𝐹𝑦) ↔ (𝑏𝑘) ∈ (𝐹𝑘)))
2827rspccva 3412 . . . . . . . . . . . . . . . . 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 20827 . . . . . . . . . . . . . . . 16 (((𝐹𝑘) ∈ Top ∧ (𝑎𝑘) ∈ (𝐹𝑘) ∧ (𝑏𝑘) ∈ (𝐹𝑘)) → ((𝑎𝑘) ∩ (𝑏𝑘)) ∈ (𝐹𝑘))
3120, 25, 29, 30syl3anc 1439 . . . . . . . . . . . . . . 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 3884 . . . . . . . . . . . . . . . . . . . 20 𝑐 ⊆ (𝑐𝑑)
34 sscon 3852 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐))
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑐)
3635sseli 3705 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑐))
3722unieqd 4554 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
3812, 37eqeq12d 2739 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑎𝑦) = (𝐹𝑦) ↔ (𝑎𝑘) = (𝐹𝑘)))
3938rspccva 3412 . . . . . . . . . . . . . . . . . 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 3885 . . . . . . . . . . . . . . . . . . . 20 𝑑 ⊆ (𝑐𝑑)
43 sscon 3852 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ (𝑐𝑑) → (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∖ (𝑐𝑑)) ⊆ (𝐴𝑑)
4544sseli 3705 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝐴 ∖ (𝑐𝑑)) → 𝑘 ∈ (𝐴𝑑))
4613, 37eqeq12d 2739 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑘 → ((𝑏𝑦) = (𝐹𝑦) ↔ (𝑏𝑘) = (𝐹𝑘)))
4746rspccva 3412 . . . . . . . . . . . . . . . . . 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 3923 . . . . . . . . . . . . . . . 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 3930 . . . . . . . . . . . . . . . 16 ( (𝐹𝑘) ∩ (𝐹𝑘)) = (𝐹𝑘)
5149, 50syl6eq 2774 . . . . . . . . . . . . . . 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 21500 . . . . . . . . . . . . . 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 2807 . . . . . . . . . . . . 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 3140 . . . . . . . . . . 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 1113 . . . . . . . . 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 3917 . . . . . . . . 9 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦)))
61 ixpin 8050 . . . . . . . . 9 X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)) = (X𝑦𝐴 (𝑎𝑦) ∩ X𝑦𝐴 (𝑏𝑦))
6260, 61syl6eqr 2776 . . . . . . . 8 ((𝑋 = X𝑦𝐴 (𝑎𝑦) ∧ 𝑌 = X𝑦𝐴 (𝑏𝑦)) → (𝑋𝑌) = X𝑦𝐴 ((𝑎𝑦) ∩ (𝑏𝑦)))
6362eleq1d 2788 . . . . . . 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 1975 . . 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 1596  wex 1817  wcel 2103  {cab 2710  wral 3014  wrex 3015  cdif 3677  cun 3678  cin 3679  wss 3680   cuni 4544   Fn wfn 5996  wf 5997  cfv 6001  Xcixp 8025  Fincfn 8072  Topctop 20821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-oadd 7684  df-er 7862  df-ixp 8026  df-en 8073  df-fin 8076  df-top 20822
This theorem is referenced by:  ptbasin2  21504
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