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Theorem ptbasfi 23501

Description: The basis for the product topology can also be written as the set of finite intersections of "cylinder sets", the preimages of projections into one factor from open sets in the factor. (We have to add 𝑋 itself to the list because if 𝐴 is empty we get (fi‘∅) = ∅ while 𝐵 = {∅}.) (Contributed by Mario Carneiro, 3-Feb-2015.)

**Hypotheses**
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))}
ptbasfi.2	⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)

**Assertion**
Ref	Expression
ptbasfi	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))

Distinct variable groups: 𝑘,𝑛,𝑢,𝐵 𝑤,𝑔,𝑥,𝑦,𝑛,𝑘,𝑢,𝑧,𝐴 𝑔,𝐹,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧 𝑔,𝑋,𝑘,𝑢,𝑤,𝑥,𝑧 𝑔,𝑉,𝑘,𝑛,𝑢,𝑤,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐵(𝑥,𝑦,𝑧,𝑤,𝑔) 𝑋(𝑦,𝑛)

Proof of Theorem ptbasfi Dummy variables 𝑠 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptbas.1	. . . . 5 ⊢ 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 23492	. . . 4 ⊢ (𝑠 ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))
3		df-3an 1088	. . . . . . . 8 ⊢ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ↔ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
4		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))
5		disjdif2 4439	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (𝐴 ∖ 𝑚) = 𝐴)
6	5	raleqdv 3296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
7	6	biimpac 478	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦))
8		ixpeq2 8861	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
10		ptbasfi.2	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
11		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑦 → (𝐹‘𝑛) = (𝐹‘𝑦))
12	11	unieqd 4880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑦))
13	12	cbvixpv 8865	. . . . . . . . . . . . . . . 16 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
14	10, 13	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
15	9, 14	eqtr4di 2782	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = 𝑋)
16	4, 15	sylan 580	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = 𝑋)
17		ssv 3968	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ⊆ V
18		iineq1 4969	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
19		0iin 5023	. . . . . . . . . . . . . . . . 17 ⊢ ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V
20	18, 19	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V)
21	17, 20	sseqtrrid 3987	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∩ 𝑚) = ∅ → 𝑋 ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
22	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) = ∅) → 𝑋 ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
23		dfss2 3929	. . . . . . . . . . . . . 14 ⊢ (𝑋 ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ↔ (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = 𝑋)
24	22, 23	sylib 218	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) = ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = 𝑋)
25	16, 24	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))))
26		simplll 774	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top))
27		inss1 4196	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝐴
28		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ (𝐴 ∩ 𝑚))
29	27, 28	sselid 3941	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ 𝐴)
30		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (ℎ‘𝑦) = (ℎ‘𝑛))
31		fveq2 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
32	30, 31	eleq12d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑛 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
33		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
34	33	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
35	32, 34, 29	rspcdva 3586	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (ℎ‘𝑛) ∈ (𝐹‘𝑛))
36	14	ptpjpre1 23491	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑛 ∈ 𝐴 ∧ (ℎ‘𝑛) ∈ (𝐹‘𝑛))) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
37	26, 29, 35, 36	syl12anc 836	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
38	37	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
39	38	iineq2dv 4977	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
40		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → (𝐴 ∩ 𝑚) ≠ ∅)
41		cnvimass 6042	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
42		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
43	42	dmmptss 6202	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ⊆ 𝑋
44	41, 43	sstri 3953	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋
45	44, 14	sseqtri 3992	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
46	45	rgenw 3048	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
47		r19.2z 4454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
48	40, 46, 47	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
49		iinss 5015	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
50	48, 49	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
51	50, 14	sseqtrrdi 3985	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋)
52		sseqin2 4182	. . . . . . . . . . . . . 14 ⊢ (∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋 ↔ (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
53	51, 52	sylib 218	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
54	33	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
55		ssralv 4012	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ∈ (𝐹‘𝑦)))
56	27, 55	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ∈ (𝐹‘𝑦))
57		elssuni 4897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦))
58		iffalse 4493	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
59	58	sseq2d 3976	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑦 = 𝑛 → ((ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦)))
60	57, 59	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (¬ 𝑦 = 𝑛 → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
61		ssid 3966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)
62		iftrue 4490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
63	62, 30	eqtr4d 2767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
64	61, 63	sseqtrrid 3987	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑛 → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
65	60, 64	pm2.61d2 181	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
66	65	ralrimivw 3129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑛 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
67		ssiin 5014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ‘𝑦) ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
68	66, 67	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
69	68	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
70	62	equcoms 2020	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
71		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → (ℎ‘𝑛) = (ℎ‘𝑦))
72	70, 71	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
73	72	sseq1d 3975	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦) ↔ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)))
74	73	rspcev 3585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
75	61, 74	mpan2 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
76		iinss 5015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
77	75, 76	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
78	77	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
79	69, 78	eqssd 3961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
80	79	ralimiaa 3065	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
81	54, 56, 80	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
82		eldifn 4091	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑚) → ¬ 𝑦 ∈ 𝑚)
83	82	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 ∈ 𝑚)
84		inss2 4197	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝑚
85		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ (𝐴 ∩ 𝑚))
86	84, 85	sselid 3941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ 𝑚)
87		eleq1 2816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ 𝑚 ↔ 𝑛 ∈ 𝑚))
88	86, 87	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑦 = 𝑛 → 𝑦 ∈ 𝑚))
89	83, 88	mtod 198	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 = 𝑛)
90	89, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
91	90	iineq2dv 4977	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦))
92		iinconst 4962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∩ 𝑚) ≠ ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑦))
93	92	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑦))
94	91, 93	eqtr2d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∪ (𝐹‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
95		eqeq1 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ‘𝑦) = ∪ (𝐹‘𝑦) → ((ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ ∪ (𝐹‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
96	94, 95	syl5ibrcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ((ℎ‘𝑦) = ∪ (𝐹‘𝑦) → (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
97	96	ralimdva 3145	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) ≠ ∅ → (∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) → ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
98	4, 97	mpan9 506	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
99		inundif 4438	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚)) = 𝐴
100	99	raleqi 3294	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚))(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
101		ralunb 4156	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚))(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ (∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
102	100, 101	bitr3i 277	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ (∀𝑦 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
103	81, 98, 102	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
104		ixpeq2 8861	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = X𝑦 ∈ 𝐴 ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
105	103, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = X𝑦 ∈ 𝐴 ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
106		ixpiin 8874	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∩ 𝑚) ≠ ∅ → X𝑦 ∈ 𝐴 ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
107	106	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → X𝑦 ∈ 𝐴 ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
108	105, 107	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)X𝑦 ∈ 𝐴 if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
109	39, 53, 108	3eqtr4rd 2775	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝐴 ∩ 𝑚) ≠ ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))))
110	25, 109	pm2.61dane 3012	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))))
111		ixpexg 8872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V)
112		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹‘𝑛) ∈ V
113	112	uniex 7697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∪ (𝐹‘𝑛) ∈ V
114	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ 𝐴 → ∪ (𝐹‘𝑛) ∈ V)
115	111, 114	mprg 3050	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V
116	10, 115	eqeltri 2824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ V
117	116	mptex 7179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
118	117	cnvex 7881	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
119	118	imaex 7870	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V
120	119	dfiin2 4993	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))}
121		inteq 4909	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∩ ∅)
122	120, 121	eqtrid 2776	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ ∅)
123		int0 4922	. . . . . . . . . . . . . . . . 17 ⊢ ∩ ∅ = V
124	122, 123	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V)
125	124	ineq2d 4179	. . . . . . . . . . . . . . 15 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = (𝑋 ∩ V))
126		inv1 4357	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∩ V) = 𝑋
127	125, 126	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = 𝑋)
128	127	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = 𝑋)
129		snex 5386	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑋} ∈ V
130	1	ptbas 23499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
131	1, 10	ptpjpre2 23500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
132	131	ralrimivva 3178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
133		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
134	133	fmpox 8025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵 ↔ (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)):∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝐵)
135	132, 134	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)):∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝐵)
136	135	frnd 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝐵)
137	130, 136	ssexd 5274	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
138		unexg 7699	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} ∈ V ∧ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
139	129, 137, 138	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
140		ssfii 9346	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
141	139, 140	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
142	141	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
143		ssun1 4137	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
144	116	snss 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ↔ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
145	143, 144	mpbir 231	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ∈ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
146	145	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → 𝑋 ∈ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
147	142, 146	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
148	147	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅) → 𝑋 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
149	128, 148	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
150	139	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
151		nfv 1914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
152		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛𝐴
153		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(𝐹‘𝑘)
154		nfixp1 8868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑛X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
155	10, 154	nfcxfr 2889	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛𝑋
156		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛(𝑤‘𝑘)
157	155, 156	nfmpt 5200	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑛(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
158	157	nfcnv 5832	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
159		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛𝑢
160	158, 159	nfima 6028	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)
161	152, 153, 160	nfmpo 7451	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
162	161	nfrn 5905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑛ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
163	162	nfcri 2883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
164		df-ov 7372	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))(ℎ‘𝑛)) = ((𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))‘⟨𝑛, (ℎ‘𝑛)⟩)
165	119	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V)
166		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑛 → (𝑤‘𝑘) = (𝑤‘𝑛))
167	166	mpteq2dv 5196	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑛 → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
168	167	cnveqd 5829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑛 → ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
169	168	imaeq1d 6019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑛 → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢))
170		imaeq2 6016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (ℎ‘𝑛) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
171	169, 170	sylan9eq 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 = 𝑛 ∧ 𝑢 = (ℎ‘𝑛)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
172		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
173	171, 172, 133	ovmpox 7522	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ 𝐴 ∧ (ℎ‘𝑛) ∈ (𝐹‘𝑛) ∧ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V) → (𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))(ℎ‘𝑛)) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
174	29, 35, 165, 173	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))(ℎ‘𝑛)) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
175	164, 174	eqtr3id 2778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ((𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))‘⟨𝑛, (ℎ‘𝑛)⟩) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
176	135	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)):∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))⟶𝐵)
177	176	ffnd 6671	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) Fn ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘)))
178		opeliunxp 5698	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) ↔ (𝑛 ∈ 𝐴 ∧ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
179	29, 35, 178	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)))
180		sneq 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
181		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
182	180, 181	xpeq12d 5662	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑘 → ({𝑛} × (𝐹‘𝑛)) = ({𝑘} × (𝐹‘𝑘)))
183	182	cbviunv 4999	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) = ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))
184	179, 183	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘)))
185		fnfvelrn 7034	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) Fn ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘)) ∧ ⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))) → ((𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))‘⟨𝑛, (ℎ‘𝑛)⟩) ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
186	177, 184, 185	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ((𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))‘⟨𝑛, (ℎ‘𝑛)⟩) ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
187	175, 186	eqeltrrd 2829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
188		eleq1 2816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) → (𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ↔ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
189	187, 188	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) → 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
190	189	ex 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → (𝑛 ∈ (𝐴 ∩ 𝑚) → (𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) → 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
191	151, 163, 190	rexlimd 3242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → (∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) → 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
192	191	abssdv 4028	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ⊆ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
193		ssun2 4138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
194	192, 193	sstrdi 3956	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
195	194	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
196		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅)
197		simplrl 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → 𝑚 ∈ Fin)
198		ssfi 9114	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ Fin ∧ (𝐴 ∩ 𝑚) ⊆ 𝑚) → (𝐴 ∩ 𝑚) ∈ Fin)
199	197, 84, 198	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → (𝐴 ∩ 𝑚) ∈ Fin)
200		abrexfi 9279	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ 𝑚) ∈ Fin → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ Fin)
201	199, 200	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ Fin)
202		elfir 9342	. . . . . . . . . . . . . . . . . 18 ⊢ ((({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V ∧ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅ ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ Fin)) → ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
203	150, 195, 196, 201, 202	syl13anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
204	120, 203	eqeltrid 2832	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
205		elssuni 4897	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
206	204, 205	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
207		fiuni 9355	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
208	139, 207	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
209	116	pwid 4581	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ 𝒫 𝑋
210	209	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211	210	snssd 4769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
212	1	ptuni2 23496	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐵)
213	10, 212	eqtrid 2776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 = ∪ 𝐵)
214		eqimss2 4003	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋)
215	213, 214	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ 𝑋)
216		sspwuni 5059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋)
217	215, 216	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218	136, 217	sstrd 3954	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219	211, 218	unssd 4151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220		sspwuni 5059	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋 ↔ ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝑋)
221	219, 220	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝑋)
222		elssuni 4897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 ∈ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑋 ⊆ ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
223	145, 222	mp1i 13	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ⊆ ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
224	221, 223	eqssd 3961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = 𝑋)
225	208, 224	eqtr3d 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) = 𝑋)
226	225	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) = 𝑋)
227	206, 226	sseqtrd 3980	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋)
228	227, 52	sylib 218	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
229	228, 204	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) ∧ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ≠ ∅) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
230	149, 229	pm2.61dane 3012	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → (𝑋 ∩ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
231	110, 230	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) ∧ (𝑚 ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
232	231	rexlimdvaa 3135	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) → (∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
233	232	impr 454	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦)) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
234	3, 233	sylan2b 594	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
235		eleq1 2816	. . . . . . 7 ⊢ (𝑠 = X𝑦 ∈ 𝐴 (ℎ‘𝑦) → (𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ↔ X𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
236	234, 235	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) → (𝑠 = X𝑦 ∈ 𝐴 (ℎ‘𝑦) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
237	236	expimpd 453	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
238	237	exlimdv 1933	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑚 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)) → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
239	2, 238	biimtrid 242	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (𝑠 ∈ 𝐵 → 𝑠 ∈ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))))
240	239	ssrdv 3949	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
241	1	ptbasid 23495	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ 𝐵)
242	10, 241	eqeltrid 2832	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝐵)
243	242	snssd 4769	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244	243, 136	unssd 4151	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝐵)
245		fiss 9351	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝐵) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
246	130, 244, 245	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
247	1	ptbasin2 23498	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248	246, 247	sseqtrd 3980	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ 𝐵)
249	240, 248	eqssd 3961	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 = (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 {cab 2707 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3444 ∖ cdif 3908 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 ifcif 4484 𝒫 cpw 4559 {csn 4585 ⟨cop 4591 ∪ cuni 4867 ∩ cint 4906 ∪ ciun 4951 ∩ ciin 4952 ↦ cmpt 5183 × cxp 5629 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 Xcixp 8847 Fincfn 8895 ficfi 9337 Topctop 22813 TopBasesctb 22865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-1o 8411 df-2o 8412 df-ixp 8848 df-en 8896 df-dom 8897 df-fin 8899 df-fi 9338 df-top 22814 df-bases 22866

This theorem is referenced by: ptval2 23521 xkoptsub 23574 ptcmplem1 23972 prdsxmslem2 24450

W3C validator