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Theorem ptbasfi 23791

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 23782	. . . 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 1105	. . . . . . . 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 785	. . . . . . . . . . . . . 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 4443	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (𝐴 ∖ 𝑚) = 𝐴)
6	5	raleqdv 3325	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦)))
7	6	biimpac 484	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦))
8		ixpeq2 8915	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐴 (ℎ‘𝑦) = ∪ (𝐹‘𝑦) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦))
9	7, 8	syl 18	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑦 → (𝐹‘𝑛) = (𝐹‘𝑦))
12	11	unieqd 4887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑦))
13	12	cbvixpv 8919	. . . . . . . . . . . . . . . 16 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
14	10, 13	eqtri 2788	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
15	9, 14	eqtr4di 2818	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = 𝑋)
16	4, 15	sylan 592	. . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ⊆ V
18		iineq1 4976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
19		0iin 5030	. . . . . . . . . . . . . . . . 17 ⊢ ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V
20	18, 19	eqtrdi 2816	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V)
21	17, 20	sseqtrrid 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∩ 𝑚) = ∅ → 𝑋 ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
22	21	adantl 487	. . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 2803	. . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝐴
28		simpr 490	. . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (ℎ‘𝑦) = (ℎ‘𝑛))
31		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
32	30, 31	eleq12d 2859	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑛 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
33		simprr 785	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
34	33	ad2antrr 739	. . . . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . 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 23781	. . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 4984	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 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 6086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
42		eqid 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
43	42	dmmptss 6244	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ⊆ 𝑋
44	41, 43	sstri 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋
45	44, 14	sseqtri 3986	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
46	45	rgenw 3085	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
47		r19.2z 4462	. . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . 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 5023	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 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 4007	. . . . . . . . . . . . . . . . . 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 4906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦))
58		iffalse 4498	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
59	58	sseq2d 3970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑦 = 𝑛 → ((ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦)))
60	57, 59	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (¬ 𝑦 = 𝑛 → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
61		ssid 3960	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)
62		iftrue 4495	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
63	62, 30	eqtr4d 2803	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
64	61, 63	sseqtrrid 3981	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑛 → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
65	60, 64	pm2.61d2 183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
66	65	ralrimivw 3163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑛 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
67		ssiin 5022	. . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
69	68	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) ⊆ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
70	62	equcoms 2053	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
71		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → (ℎ‘𝑛) = (ℎ‘𝑦))
72	70, 71	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
73	72	sseq1d 3969	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦) ↔ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)))
74	73	rspcev 3583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
75	61, 74	mpan2 704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
76		iinss 5023	. . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
78	77	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
79	69, 78	eqssd 3955	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
80	79	ralimiaa 3103	. . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑚) → ¬ 𝑦 ∈ 𝑚)
83	82	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 ∈ 𝑚)
84		inss2 4190	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝑚
85		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ (𝐴 ∩ 𝑚))
86	84, 85	sselid 3936	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ 𝑚)
87		eleq1 2853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ 𝑚 ↔ 𝑛 ∈ 𝑚))
88	86, 87	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑦 = 𝑛 → 𝑦 ∈ 𝑚))
89	83, 88	mtod 201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 = 𝑛)
90	89, 58	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
91	90	iineq2dv 4984	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦))
92		iinconst 4969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∩ 𝑚) ≠ ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑦))
93	92	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦) = ∪ (𝐹‘𝑦))
94	91, 93	eqtr2d 2801	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∪ (𝐹‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
95		eqeq1 2769	. . . . . . . . . . . . . . . . . . 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 250	. . . . . . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . . . . 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 516	. . . . . . . . . . . . . . . 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 4442	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚)) = 𝐴
100	99	raleqi 3323	. . . . . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 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 8915	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 8928	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 2811	. . . . . . . . . . . 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 3047	. . . . . . . . . . 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 8926	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V)
112		fvex 6898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹‘𝑛) ∈ V
113	112	uniex 7749	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∪ (𝐹‘𝑛) ∈ V
114	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ 𝐴 → ∪ (𝐹‘𝑛) ∈ V)
115	111, 114	mprg 3087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V
116	10, 115	eqeltri 2861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ V
117	116	mptex 7228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
118	117	cnvex 7928	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
119	118	imaex 7917	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V
120	119	dfiin2 4999	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))}
121		inteq 4917	. . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ ∅)
123		int0 4929	. . . . . . . . . . . . . . . . 17 ⊢ ∩ ∅ = V
124	122, 123	eqtrdi 2816	. . . . . . . . . . . . . . . 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 4173	. . . . . . . . . . . . . . 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 4355	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∩ V) = 𝑋
127	125, 126	eqtrdi 2816	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 5412	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑋} ∈ V
130	1	ptbas 23789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
131	1, 10	ptpjpre2 23790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
132	131	ralrimivva 3210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
133		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
134	133	fmpox 8070	. . . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝐵)
137	130, 136	ssexd 5297	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
138		unexg 7751	. . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
140		ssfii 9386	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
144	116	snss 4752	. . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . . . . . 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 2927	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛𝐴
153		nfcv 2927	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(𝐹‘𝑘)
154		nfixp1 8922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑛X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
155	10, 154	nfcxfr 2925	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛𝑋
156		nfcv 2927	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛(𝑤‘𝑘)
157	155, 156	nfmpt 5211	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑛(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
158	157	nfcnv 5866	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
159		nfcv 2927	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛𝑢
160	158, 159	nfima 6072	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)
161	152, 153, 160	nfmpo 7501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
162	161	nfrn 5944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑛ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
163	162	nfcri 2919	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
164		df-ov 7422	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑛 → (𝑤‘𝑘) = (𝑤‘𝑛))
167	166	mpteq2dv 5207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑛 → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
168	167	cnveqd 5863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑛 → ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
169	168	imaeq1d 6063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑛 → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢))
170		imaeq2 6060	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (ℎ‘𝑛) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
171	169, 170	sylan9eq 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 = 𝑛 ∧ 𝑢 = (ℎ‘𝑛)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
172		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
173	171, 172, 133	ovmpox 7572	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6710	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) ↔ (𝑛 ∈ 𝐴 ∧ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
179	29, 35, 178	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
181		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
182	180, 181	xpeq12d 5694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑘 → ({𝑛} × (𝐹‘𝑛)) = ({𝑘} × (𝐹‘𝑘)))
183	182	cbviunv 5005	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) = ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))
184	179, 183	eleqtrdi 2875	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . . . . . . . . . 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 250	. . . . . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . . . . 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 3274	. . . . . . . . . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . . . . . . . . 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 4132	. . . . . . . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . 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 9164	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ Fin ∧ (𝐴 ∩ 𝑚) ⊆ 𝑚) → (𝐴 ∩ 𝑚) ∈ Fin)
199	197, 84, 198	sylancl 598	. . . . . . . . . . . . . . . . . . 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 9316	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ 𝑚) ∈ Fin → {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} ∈ Fin)
201	199, 200	syl 18	. . . . . . . . . . . . . . . . . 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 9382	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . . 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 4906	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 9395	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . 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 4587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ 𝒫 𝑋
210	209	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211	210	snssd 4754	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
212	1	ptuni2 23786	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐵)
213	10, 212	eqtrid 2812	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 = ∪ 𝐵)
214		eqimss2 3997	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋)
215	213, 214	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ 𝑋)
216		sspwuni 5068	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋)
217	215, 216	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218	136, 217	sstrd 3948	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219	211, 218	unssd 4145	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220		sspwuni 5068	. . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝑋)
222		elssuni 4906	. . . . . . . . . . . . . . . . . . 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 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ⊆ ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))))
224	221, 223	eqssd 3955	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = 𝑋)
225	208, 224	eqtr3d 2802	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) = 𝑋)
226	225	ad3antrrr 743	. . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 3047	. . . . . . . . . . 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 2865	. . . . . . . . . 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 3169	. . . . . . . . 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 460	. . . . . . . 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 606	. . . . . . 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 2853	. . . . . . 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 250	. . . . . 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 459	. . . . 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 1966	. . . 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 245	. . 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 3944	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
241	1	ptbasid 23785	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ 𝐵)
242	10, 241	eqeltrid 2869	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝐵)
243	242	snssd 4754	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244	243, 136	unssd 4145	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝐵)
245		fiss 9391	. . . 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 596	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
247	1	ptbasin2 23788	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248	246, 247	sseqtrd 3974	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ 𝐵)
249	240, 248	eqssd 3955	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 {cab 2743 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ifcif 4489 𝒫 cpw 4564 {csn 4591 ⟨cop 4597 ∪ cuni 4874 ∩ cint 4914 ∪ ciun 4958 ∩ ciin 4959 ↦ cmpt 5194 × cxp 5661 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 “ cima 5666 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 Xcixp 8901 Fincfn 8949 ficfi 9377 Topctop 23102 TopBasesctb 23154

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 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 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-1o 8459 df-2o 8460 df-ixp 8902 df-en 8950 df-dom 8951 df-fin 8953 df-fi 9378 df-top 23103 df-bases 23155

This theorem is used by: ptval2 23811 xkoptsub 23864 ptcmplem1 24262 prdsxmslem2 24739

W3C validator