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Theorem ptbasin 23560

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 23555	. . . . 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 23555	. . . . 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 634	. . . 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 1962	. . . 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 279	. . 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 662	. . . . 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 1453	. . . . . . . . 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 1094	. . . . . . . . 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 276	. . . . . . . 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 3211	. . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑎‘𝑦) = (𝑎‘𝑘))
13		fveq2 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑘 → (𝑏‘𝑦) = (𝑏‘𝑘))
14	12, 13	ineq12d 4150	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = ((𝑎‘𝑘) ∩ (𝑏‘𝑘)))
15	14	cbvixpv 8853	. . . . . . . . . . . . . 14 ⊢ X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = X𝑘 ∈ 𝐴 ((𝑎‘𝑘) ∩ (𝑏‘𝑘))
16		simpl1l 1231	. . . . . . . . . . . . . . 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 9095	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ Fin ∧ 𝑑 ∈ Fin) → (𝑐 ∪ 𝑑) ∈ Fin)
18	17	ad2antrl 734	. . . . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . 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 1235	. . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
23	12, 22	eleq12d 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝑎‘𝑘) ∈ (𝐹‘𝑘)))
24	23	rspccva 3559	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝐴 (𝑎‘𝑦) ∈ (𝐹‘𝑦) ∧ 𝑘 ∈ 𝐴) → (𝑎‘𝑘) ∈ (𝐹‘𝑘))
25	21, 24	sylan 586	. . . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑘 → ((𝑏‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝑏‘𝑘) ∈ (𝐹‘𝑘)))
28	27	rspccva 3559	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝐴 (𝑏‘𝑦) ∈ (𝐹‘𝑦) ∧ 𝑘 ∈ 𝐴) → (𝑏‘𝑘) ∈ (𝐹‘𝑘))
29	26, 28	sylan 586	. . . . . . . . . . . . . . . 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 22882	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑘) ∈ Top ∧ (𝑎‘𝑘) ∈ (𝐹‘𝑘) ∧ (𝑏‘𝑘) ∈ (𝐹‘𝑘)) → ((𝑎‘𝑘) ∩ (𝑏‘𝑘)) ∈ (𝐹‘𝑘))
31	20, 25, 29, 30	syl3anc 1379	. . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . 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 4107	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑐 ⊆ (𝑐 ∪ 𝑑)
34		sscon 4073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ⊆ (𝑐 ∪ 𝑑) → (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑐))
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑐)
36	35	sseli 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑)) → 𝑘 ∈ (𝐴 ∖ 𝑐))
37	22	unieqd 4851	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑘 → ∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑘))
38	12, 37	eqeq12d 2755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → ((𝑎‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝑎‘𝑘) = ∪ (𝐹‘𝑘)))
39	38	rspccva 3559	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑐)(𝑎‘𝑦) = ∪ (𝐹‘𝑦) ∧ 𝑘 ∈ (𝐴 ∖ 𝑐)) → (𝑎‘𝑘) = ∪ (𝐹‘𝑘))
40	32, 36, 39	syl2an 602	. . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . 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 4108	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑑 ⊆ (𝑐 ∪ 𝑑)
43		sscon 4073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ⊆ (𝑐 ∪ 𝑑) → (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑑))
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∖ (𝑐 ∪ 𝑑)) ⊆ (𝐴 ∖ 𝑑)
45	44	sseli 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝐴 ∖ (𝑐 ∪ 𝑑)) → 𝑘 ∈ (𝐴 ∖ 𝑑))
46	13, 37	eqeq12d 2755	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑘 → ((𝑏‘𝑦) = ∪ (𝐹‘𝑦) ↔ (𝑏‘𝑘) = ∪ (𝐹‘𝑘)))
47	46	rspccva 3559	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑑)(𝑏‘𝑦) = ∪ (𝐹‘𝑦) ∧ 𝑘 ∈ (𝐴 ∖ 𝑑)) → (𝑏‘𝑘) = ∪ (𝐹‘𝑘))
48	41, 45, 47	syl2an 602	. . . . . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . 16 ⊢ (∪ (𝐹‘𝑘) ∩ ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)
51	49, 50	eqtrdi 2790	. . . . . . . . . . . . . . 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 23557	. . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . 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 457	. . . . . . . . . . . 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 3196	. . . . . . . . . . 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 244	. . . . . . . . . 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 1126	. . . . . . . . 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 455	. . . . . . . 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 600	. . . . . . 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 4144	. . . . . . . . 9 ⊢ ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → (𝑋 ∩ 𝑌) = (X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∩ X𝑦 ∈ 𝐴 (𝑏‘𝑦)))
61		ixpin 8861	. . . . . . . . 9 ⊢ X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)) = (X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∩ X𝑦 ∈ 𝐴 (𝑏‘𝑦))
62	60, 61	eqtr4di 2792	. . . . . . . 8 ⊢ ((𝑋 = X𝑦 ∈ 𝐴 (𝑎‘𝑦) ∧ 𝑌 = X𝑦 ∈ 𝐴 (𝑏‘𝑦)) → (𝑋 ∩ 𝑌) = X𝑦 ∈ 𝐴 ((𝑎‘𝑦) ∩ (𝑏‘𝑦)))
63	62	eleq1d 2824	. . . . . . 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 248	. . . . . 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 454	. . . . 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 243	. . . 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 1941	. . 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 243	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∩ 𝑌) ∈ 𝐵))
69	68	imp 407	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∩ 𝑌) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 {cab 2717 ∀wral 3053 ∃wrex 3063 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∪ cuni 4838 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 Xcixp 8835 Fincfn 8883 Topctop 22876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-om 7807 df-ixp 8836 df-en 8884 df-fin 8887 df-top 22877

This theorem is referenced by: ptbasin2 23561

W3C validator