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Theorem ptbasfi 23546

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 23537	. . . 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 1089	. . . . . . . 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 773	. . . . . . . . . . . . . 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 4420	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (𝐴 ∖ 𝑚) = 𝐴)
6	5	raleqdv 3295	. . . . . . . . . . . . . . . . 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 8859	. . . . . . . . . . . . . . . 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 4863	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑦))
13	12	cbvixpv 8863	. . . . . . . . . . . . . . . 16 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
14	10, 13	eqtri 2759	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
15	9, 14	eqtr4di 2789	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = 𝑋)
16	4, 15	sylan 581	. . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ⊆ V
18		iineq1 4951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
19		0iin 5006	. . . . . . . . . . . . . . . . 17 ⊢ ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V
20	18, 19	eqtrdi 2787	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V)
21	17, 20	sseqtrrid 3965	. . . . . . . . . . . . . . 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 3907	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . 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 4177	. . . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑛 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
33		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
34	33	ad2antrr 727	. . . . . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . . . 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 23536	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . 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 4959	. . . . . . . . . . . . 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 6047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
42		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
43	42	dmmptss 6205	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ⊆ 𝑋
44	41, 43	sstri 3931	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋
45	44, 14	sseqtri 3970	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
46	45	rgenw 3055	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
47		r19.2z 4439	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 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 4999	. . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . 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 4163	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . 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 3990	. . . . . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦))
58		iffalse 4475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
59	58	sseq2d 3954	. . . . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)
62		iftrue 4472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
63	62, 30	eqtr4d 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
64	61, 63	sseqtrrid 3965	. . . . . . . . . . . . . . . . . . . . . . 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 3133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑛 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
67		ssiin 4998	. . . . . . . . . . . . . . . . . . . . 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 2022	. . . . . . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
73	72	sseq1d 3953	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦) ↔ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)))
74	73	rspcev 3564	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
75	61, 74	mpan2 692	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
76		iinss 4999	. . . . . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
80	79	ralimiaa 3073	. . . . . . . . . . . . . . . . 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 4072	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑚) → ¬ 𝑦 ∈ 𝑚)
83	82	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 ∈ 𝑚)
84		inss2 4178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝑚
85		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ (𝐴 ∩ 𝑚))
86	84, 85	sselid 3919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ 𝑚)
87		eleq1 2824	. . . . . . . . . . . . . . . . . . . . . . . 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 4959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦))
92		iinconst 4944	. . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∪ (𝐹‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
95		eqeq1 2740	. . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . 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 4419	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚)) = 𝐴
100	99	raleqi 3293	. . . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 8859	. . . . . . . . . . . . . . 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 8872	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 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 3019	. . . . . . . . . . 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 8870	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V)
112		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹‘𝑛) ∈ V
113	112	uniex 7695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∪ (𝐹‘𝑛) ∈ V
114	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ 𝐴 → ∪ (𝐹‘𝑛) ∈ V)
115	111, 114	mprg 3057	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V
116	10, 115	eqeltri 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ V
117	116	mptex 7178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
118	117	cnvex 7876	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
119	118	imaex 7865	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V
120	119	dfiin2 4975	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))}
121		inteq 4892	. . . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ ∅)
123		int0 4904	. . . . . . . . . . . . . . . . 17 ⊢ ∩ ∅ = V
124	122, 123	eqtrdi 2787	. . . . . . . . . . . . . . . 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 4160	. . . . . . . . . . . . . . 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 4338	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∩ V) = 𝑋
127	125, 126	eqtrdi 2787	. . . . . . . . . . . . . 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 5381	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑋} ∈ V
130	1	ptbas 23544	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
131	1, 10	ptpjpre2 23545	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
132	131	ralrimivva 3180	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
133		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
134	133	fmpox 8020	. . . . . . . . . . . . . . . . . . . . 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 6676	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝐵)
137	130, 136	ssexd 5265	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
138		unexg 7697	. . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
140		ssfii 9332	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . 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 4118	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
144	116	snss 4728	. . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛𝐴
153		nfcv 2898	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(𝐹‘𝑘)
154		nfixp1 8866	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑛X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
155	10, 154	nfcxfr 2896	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛𝑋
156		nfcv 2898	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛(𝑤‘𝑘)
157	155, 156	nfmpt 5183	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑛(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
158	157	nfcnv 5833	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
159		nfcv 2898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛𝑢
160	158, 159	nfima 6033	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)
161	152, 153, 160	nfmpo 7449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
162	161	nfrn 5907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑛ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
163	162	nfcri 2890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
164		df-ov 7370	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑛 → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
168	167	cnveqd 5830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑛 → ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
169	168	imaeq1d 6024	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑛 → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢))
170		imaeq2 6021	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (ℎ‘𝑛) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
171	169, 170	sylan9eq 2791	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7520	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4577	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) = ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))
184	179, 183	eleqtrdi 2846	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7032	. . . . . . . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . . . . . . . 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 3244	. . . . . . . . . . . . . . . . . . . . 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 4007	. . . . . . . . . . . . . . . . . . . 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 4119	. . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . 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 9107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ Fin ∧ (𝐴 ∩ 𝑚) ⊆ 𝑚) → (𝐴 ∩ 𝑚) ∈ Fin)
199	197, 84, 198	sylancl 587	. . . . . . . . . . . . . . . . . . 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 9262	. . . . . . . . . . . . . . . . . . 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 9328	. . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . . 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 9341	. . . . . . . . . . . . . . . . . 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 4563	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ 𝒫 𝑋
210	209	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211	210	snssd 4730	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
212	1	ptuni2 23541	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐵)
213	10, 212	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 = ∪ 𝐵)
214		eqimss2 3981	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋)
215	213, 214	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ 𝑋)
216		sspwuni 5042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋)
217	215, 216	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218	136, 217	sstrd 3932	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219	211, 218	unssd 4132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220		sspwuni 5042	. . . . . . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = 𝑋)
225	208, 224	eqtr3d 2773	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) = 𝑋)
226	225	ad3antrrr 731	. . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . 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 3019	. . . . . . . . . . 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 2836	. . . . . . . . . 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 3139	. . . . . . . . 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 595	. . . . . . 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 2824	. . . . . . 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 1935	. . . 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 3927	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
241	1	ptbasid 23540	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ 𝐵)
242	10, 241	eqeltrid 2840	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝐵)
243	242	snssd 4730	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244	243, 136	unssd 4132	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝐵)
245		fiss 9337	. . . 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 585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
247	1	ptbasin2 23543	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248	246, 247	sseqtrd 3958	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ 𝐵)
249	240, 248	eqssd 3939	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 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ∖ cdif 3886 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 ifcif 4466 𝒫 cpw 4541 {csn 4567 ⟨cop 4573 ∪ cuni 4850 ∩ cint 4889 ∪ ciun 4933 ∩ ciin 4934 ↦ cmpt 5166 × cxp 5629 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 Xcixp 8845 Fincfn 8893 ficfi 9323 Topctop 22858 TopBasesctb 22910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-1o 8405 df-2o 8406 df-ixp 8846 df-en 8894 df-dom 8895 df-fin 8897 df-fi 9324 df-top 22859 df-bases 22911

This theorem is referenced by: ptval2 23566 xkoptsub 23619 ptcmplem1 24017 prdsxmslem2 24494

W3C validator