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Theorem ptbasin 23715

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 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}

**Assertion**
Ref	Expression
ptbasin	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∩ 𝑌) ∈ 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.

Step	Hyp	Ref	Expression
1		ptbas.1	. . . . . 6 ⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
2	1	elpt 23710	. . . . 5 ⊢ (𝑋 ∈ 𝐵 ↔ ∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)))
3	1	elpt 23710	. . . . 5 ⊢ (𝑌 ∈ 𝐵 ↔ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)))
4	2, 3	anbi12i 639	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ↔ (∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))))
5		exdistrv 1985	. . . 4 ⊢ (∃𝑎∃𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))) ↔ (∃𝑎((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ∃𝑏((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))))
6	4, 5	bitr4i 281	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ↔ ∃𝑎∃𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))))
7		an4 668	. . . . 5 ⊢ ((((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))) ↔ (((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))))
8		an6 1474	. . . . . . . . 9 ⊢ (((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))) ↔ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦)) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))))
9		df-3an 1105	. . . . . . . . 9 ⊢ (((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦)) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))) ↔ (((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))))
10	8, 9	bitri 278	. . . . . . . 8 ⊢ (((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))) ↔ (((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))))
11		reeanv 3237	. . . . . . . . . . 11 ⊢ (∃𝑐 ∈ Fin ∃𝑑 ∈ Fin (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ↔ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))
12		fveq2 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑎‘𝑦) = (𝑎‘𝑘))
13		fveq2 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑏‘𝑦) = (𝑏‘𝑘))
14	12, 13	ineq12d 4175	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = ((𝑎‘𝑘) ∩ (𝑏‘𝑘)))
15	14	cbvixpv 8914	. . . . . . . . . . . . . 14 ⊢ X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = X𝑘 ∈ 𝐴 ((𝑎‘𝑘) ∩ (𝑏‘𝑘))
16		simpl1l 1243	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → 𝐴 ∈ 𝑉)
17		unfi 9156	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → (𝑐 ∪ 𝑑) ∈ Fin)
18	17	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → (𝑐 ∪ 𝑑) ∈ Fin)
19		simpl1r 1244	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → 𝐹:𝐴⟶Top)
20	19	ffvelcdmda 7081	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ Top)
21		simpl3l 1247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦))
22		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
23	12, 22	eleq12d 2857	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝑎‘𝑘) ∈ (𝐹‘𝑘)))
24	23	rspccva 3581	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ 𝑘 ∈ 𝐴) → (𝑎‘𝑘) ∈ (𝐹‘𝑘))
25	21, 24	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ 𝐴) → (𝑎‘𝑘) ∈ (𝐹‘𝑘))
26		simpl3r 1248	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))
27	13, 22	eleq12d 2857	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑘 → ((𝑏‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝑏‘𝑘) ∈ (𝐹‘𝑘)))
28	27	rspccva 3581	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ 𝑘 ∈ 𝐴) → (𝑏‘𝑘) ∈ (𝐹‘𝑘))
29	26, 28	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ 𝐴) → (𝑏‘𝑘) ∈ (𝐹‘𝑘))
30		inopn 23037	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑘) ∈ Top ∧ (𝑎‘𝑘) ∈ (𝐹‘𝑘) ∧ (𝑏‘𝑘) ∈ (𝐹‘𝑘)) → ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) ∈ (𝐹‘𝑘))
31	20, 25, 29, 30	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ 𝐴) → ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) ∈ (𝐹‘𝑘))
32		simprrl 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦))
33		ssun1 4132	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ⊆ (𝑐 ∪ 𝑑)
34		sscon 4098	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ⊆ (𝑐 ∪ 𝑑) → (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑐))
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑐)
36	35	sseli 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑)) → 𝑘 ∈ (𝐴 ∖ 𝑐))
37	22	unieqd 4886	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
38	12, 37	eqeq12d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝑎‘𝑘) = ∪ (𝐹‘𝑘)))
39	38	rspccva 3581	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ 𝑘 ∈ (𝐴 ∖ 𝑐)) → (𝑎‘𝑘) = ∪ (𝐹‘𝑘))
40	32, 36, 39	syl2an 607	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑))) → (𝑎‘𝑘) = ∪ (𝐹‘𝑘))
41		simprrr 793	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))
42		ssun2 4133	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑑 ⊆ (𝑐 ∪ 𝑑)
43		sscon 4098	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ⊆ (𝑐 ∪ 𝑑) → (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑑))
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑑)
45	44	sseli 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑)) → 𝑘 ∈ (𝐴 ∖ 𝑑))
46	13, 37	eqeq12d 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → ((𝑏‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝑏‘𝑘) = ∪ (𝐹‘𝑘)))
47	46	rspccva 3581	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦) ∧ 𝑘 ∈ (𝐴 ∖ 𝑑)) → (𝑏‘𝑘) = ∪ (𝐹‘𝑘))
48	41, 45, 47	syl2an 607	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑))) → (𝑏‘𝑘) = ∪ (𝐹‘𝑘))
49	40, 48	ineq12d 4175	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑))) → ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) = (∪ (𝐹‘𝑘) ∩ ∪ (𝐹‘𝑘)))
50		inidm 4180	. . . . . . . . . . . . . . . 16 ⊢ (∪ (𝐹‘𝑘) ∩ ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)
51	49, 50	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) ∧ 𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑))) → ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) = ∪ (𝐹‘𝑘))
52	1, 16, 18, 31, 51	elptr2 23712	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → X𝑘 ∈ 𝐴 ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) ∈ 𝐵)
53	15, 52	eqeltrid 2867	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) ∧ (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵)
54	53	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑐 ∈ Fin ∧ 𝑑 ∈ Fin)) → ((∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵))
55	54	rexlimdvva 3222	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) → (∃𝑐 ∈ Fin ∃𝑑 ∈ Fin (∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵))
56	11, 55	biimtrrid 246	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) → ((∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵))
57	56	3expb 1138	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦)))) → ((∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵))
58	57	impr 459	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (((𝑎 Fn 𝐴 ∧ 𝑏 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦))) ∧ (∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵)
59	10, 58	sylan2b 605	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵)
60		ineq12 4169	. . . . . . . . 9 ⊢ ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → (𝑋 ∩ 𝑌) = (X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∩ X𝑦 ∈ 𝐴 (𝑏‘𝑦)))
61		ixpin 8922	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = (X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∩ X𝑦 ∈ 𝐴 (𝑏‘𝑦))
62	60, 61	eqtr4di 2816	. . . . . . . 8 ⊢ ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → (𝑋 ∩ 𝑌) = X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)))
63	62	eleq1d 2848	. . . . . . 7 ⊢ ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → ((𝑋 ∩ 𝑌) ∈ 𝐵 ↔ X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) ∈ 𝐵))
64	59, 63	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ ((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)))) → ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → (𝑋 ∩ 𝑌) ∈ 𝐵))
65	64	expimpd 458	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ((((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ (𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))) → (𝑋 ∩ 𝑌) ∈ 𝐵))
66	7, 65	biimtrid 245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ((((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))) → (𝑋 ∩ 𝑌) ∈ 𝐵))
67	66	exlimdvv 1964	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∃𝑎∃𝑏(((𝑎 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑐 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦)) ∧ ((𝑏 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑑 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦))) → (𝑋 ∩ 𝑌) ∈ 𝐵))
68	6, 67	biimtrid 245	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∩ 𝑌) ∈ 𝐵))
69	68	imp 411	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∩ 𝑌) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 {cab 2741 ∀wral 3079 ∃wrex 3089 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∪ cuni 4873 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 Xcixp 8896 Fincfn 8944 Topctop 23031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-om 7864 df-ixp 8897 df-en 8945 df-fin 8948 df-top 23032

This theorem is referenced by: ptbasin2 23716

W3C validator