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Theorem ptbasfi 22640

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 22631	. . . 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 1087	. . . . . . . 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 769	. . . . . . . . . . . . . 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 4410	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) = ∅ → (𝐴 ∖ 𝑚) = 𝐴)
6	5	raleqdv 3339	. . . . . . . . . . . . . . . . 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 8657	. . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑦 → (𝐹‘𝑛) = (𝐹‘𝑦))
12	11	unieqd 4850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑦))
13	12	cbvixpv 8661	. . . . . . . . . . . . . . . 16 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
14	10, 13	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
15	9, 14	eqtr4di 2797	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (𝐴 ∖ 𝑚)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ∧ (𝐴 ∩ 𝑚) = ∅) → X𝑦 ∈ 𝐴 (ℎ‘𝑦) = 𝑋)
16	4, 15	sylan 579	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 ⊆ V
18		iineq1 4938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
19		0iin 4989	. . . . . . . . . . . . . . . . 17 ⊢ ∩ 𝑛 ∈ ∅ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V
20	18, 19	eqtrdi 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑚) = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = V)
21	17, 20	sseqtrrid 3970	. . . . . . . . . . . . . . 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		df-ss 3900	. . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 4159	. . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (ℎ‘𝑦) = (ℎ‘𝑛))
31		fveq2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
32	30, 31	eleq12d 2833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑛 → ((ℎ‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
33		simprr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))
34	33	ad2antrr 722	. . . . . . . . . . . . . . . . 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 3554	. . . . . . . . . . . . . . . 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 22630	. . . . . . . . . . . . . . . 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 833	. . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . 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 4946	. . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
42		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛))
43	42	dmmptss 6133	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ⊆ 𝑋
44	41, 43	sstri 3926	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ 𝑋
45	44, 14	sseqtri 3953	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
46	45	rgenw 3075	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ⊆ X𝑦 ∈ 𝐴 ∪ (𝐹‘𝑦)
47		r19.2z 4422	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 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 4982	. . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . . 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 4868	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦))
58		iffalse 4465	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∪ (𝐹‘𝑦))
59	58	sseq2d 3949	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑦 = 𝑛 → ((ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ↔ (ℎ‘𝑦) ⊆ ∪ (𝐹‘𝑦)))
60	57, 59	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → (¬ 𝑦 = 𝑛 → (ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦))))
61		ssid 3939	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)
62		iftrue 4462	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
63	62, 30	eqtr4d 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
64	61, 63	sseqtrrid 3970	. . . . . . . . . . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑛 ∈ (𝐴 ∩ 𝑚)(ℎ‘𝑦) ⊆ if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
67		ssiin 4981	. . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . 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 2024	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑛))
71		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑦 → (ℎ‘𝑛) = (ℎ‘𝑦))
72	70, 71	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑦 → if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = (ℎ‘𝑦))
73	72	sseq1d 3948	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑦 → (if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦) ↔ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)))
74	73	rspcev 3552	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ⊆ (ℎ‘𝑦)) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
75	61, 74	mpan2 687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (𝐴 ∩ 𝑚) → ∃𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) ⊆ (ℎ‘𝑦))
76		iinss 4982	. . . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝐴 ∩ 𝑚) ∧ (ℎ‘𝑦) ∈ (𝐹‘𝑦)) → (ℎ‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
80	79	ralimiaa 3085	. . . . . . . . . . . . . . . . 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 4058	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∖ 𝑚) → ¬ 𝑦 ∈ 𝑚)
83	82	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → ¬ 𝑦 ∈ 𝑚)
84		inss2 4160	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝑚) ⊆ 𝑚
85		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ (𝐴 ∩ 𝑚))
86	84, 85	sselid 3915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → 𝑛 ∈ 𝑚)
87		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑛 → (𝑦 ∈ 𝑚 ↔ 𝑛 ∈ 𝑚))
88	86, 87	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) ∧ 𝑛 ∈ (𝐴 ∩ 𝑚)) → (𝑦 = 𝑛 → 𝑦 ∈ 𝑚))
89	83, 88	mtod 197	. . . . . . . . . . . . . . . . . . . . . 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 4946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)∪ (𝐹‘𝑦))
92		iinconst 4931	. . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∩ 𝑚) ≠ ∅ ∧ 𝑦 ∈ (𝐴 ∖ 𝑚)) → ∪ (𝐹‘𝑦) = ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)if(𝑦 = 𝑛, (ℎ‘𝑛), ∪ (𝐹‘𝑦)))
95		eqeq1 2742	. . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . 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 3102	. . . . . . . . . . . . . . . . 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 4409	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∩ 𝑚) ∪ (𝐴 ∖ 𝑚)) = 𝐴
100	99	raleqi 3337	. . . . . . . . . . . . . . . . 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 4121	. . . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . 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 8657	. . . . . . . . . . . . . . 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 8670	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 2789	. . . . . . . . . . . 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 3031	. . . . . . . . . . 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 8668	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V)
112		fvex 6769	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹‘𝑛) ∈ V
113	112	uniex 7572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∪ (𝐹‘𝑛) ∈ V
114	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ 𝐴 → ∪ (𝐹‘𝑛) ∈ V)
115	111, 114	mprg 3077	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ V
116	10, 115	eqeltri 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ V
117	116	mptex 7081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
118	117	cnvex 7746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) ∈ V
119	118	imaex 7737	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) ∈ V
120	119	dfiin2 4960	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ {𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))}
121		inteq 4879	. . . . . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧 ∣ ∃𝑛 ∈ (𝐴 ∩ 𝑚)𝑧 = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛))} = ∅ → ∩ 𝑛 ∈ (𝐴 ∩ 𝑚)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)) = ∩ ∅)
123		int0 4890	. . . . . . . . . . . . . . . . 17 ⊢ ∩ ∅ = V
124	122, 123	eqtrdi 2795	. . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . 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 4325	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∩ V) = 𝑋
127	125, 126	eqtrdi 2795	. . . . . . . . . . . . . 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 5349	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑋} ∈ V
130	1	ptbas 22638	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ TopBases)
131	1, 10	ptpjpre2 22639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘))) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
132	131	ralrimivva 3114	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑘 ∈ 𝐴 ∀𝑢 ∈ (𝐹‘𝑘)(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝐵)
133		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
134	133	fmpox 7880	. . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . 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 6592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝐵)
137	130, 136	ssexd 5243	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ∈ V)
138		unexg 7577	. . . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ∈ V)
140		ssfii 9108	. . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ⊆ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
144	116	snss 4716	. . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 1918	. . . . . . . . . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛𝐴
153		nfcv 2906	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(𝐹‘𝑘)
154		nfixp1 8664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ Ⅎ𝑛X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
155	10, 154	nfcxfr 2904	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛𝑋
156		nfcv 2906	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑛(𝑤‘𝑘)
157	155, 156	nfmpt 5177	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑛(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
158	157	nfcnv 5776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
159		nfcv 2906	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑛𝑢
160	158, 159	nfima 5966	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑛(^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)
161	152, 153, 160	nfmpo 7335	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑛(𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
162	161	nfrn 5850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑛ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
163	162	nfcri 2893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛 𝑧 ∈ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
164		df-ov 7258	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 = 𝑛 → (𝑤‘𝑘) = (𝑤‘𝑛))
167	166	mpteq2dv 5172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑘 = 𝑛 → (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
168	167	cnveqd 5773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 = 𝑛 → ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = ^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)))
169	168	imaeq1d 5957	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 = 𝑛 → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢))
170		imaeq2 5954	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (ℎ‘𝑛) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
171	169, 170	sylan9eq 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 = 𝑛 ∧ 𝑢 = (ℎ‘𝑛)) → (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑛)) “ (ℎ‘𝑛)))
172		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
173	171, 172, 133	ovmpox 7404	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2793	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6585	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5645	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑛, (ℎ‘𝑛)⟩ ∈ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) ↔ (𝑛 ∈ 𝐴 ∧ (ℎ‘𝑛) ∈ (𝐹‘𝑛)))
179	29, 35, 178	sylanbrc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
181		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
182	180, 181	xpeq12d 5611	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = 𝑘 → ({𝑛} × (𝐹‘𝑛)) = ({𝑘} × (𝐹‘𝑘)))
183	182	cbviunv 4966	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ∪ 𝑛 ∈ 𝐴 ({𝑛} × (𝐹‘𝑛)) = ∪ 𝑘 ∈ 𝐴 ({𝑘} × (𝐹‘𝑘))
184	179, 183	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6940	. . . . . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . . . . . . . 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 3245	. . . . . . . . . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . . . . . . . . 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 4103	. . . . . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . 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 8918	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ Fin ∧ (𝐴 ∩ 𝑚) ⊆ 𝑚) → (𝐴 ∩ 𝑚) ∈ Fin)
199	197, 84, 198	sylancl 585	. . . . . . . . . . . . . . . . . . 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 9049	. . . . . . . . . . . . . . . . . . 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 9104	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . 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 4868	. . . . . . . . . . . . . . . 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 9117	. . . . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑋 ∈ 𝒫 𝑋
210	209	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝒫 𝑋)
211	210	snssd 4739	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝒫 𝑋)
212	1	ptuni2 22635	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐵)
213	10, 212	eqtrid 2790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 = ∪ 𝐵)
214		eqimss2 3974	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋)
215	213, 214	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ⊆ 𝑋)
216		sspwuni 5025	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋)
217	215, 216	sylibr 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 𝑋)
218	136, 217	sstrd 3927	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)) ⊆ 𝒫 𝑋)
219	211, 218	unssd 4116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝒫 𝑋)
220		sspwuni 5025	. . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝑋)
222		elssuni 4868	. . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) = 𝑋)
225	208, 224	eqtr3d 2780	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) = 𝑋)
226	225	ad3antrrr 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))) ∧ {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 3957	. . . . . . . . . . . . . 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 217	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 3031	. . . . . . . . . . 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 2839	. . . . . . . . . 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 3213	. . . . . . . . 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 593	. . . . . . 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 2826	. . . . . . 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 246	. . . . . 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 1937	. . . 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	syl5bi 241	. . 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 3923	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ⊆ (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))))
241	1	ptbasid 22634	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ 𝐵)
242	10, 241	eqeltrid 2843	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝑋 ∈ 𝐵)
243	242	snssd 4739	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑋} ⊆ 𝐵)
244	243, 136	unssd 4116	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) ⊆ 𝐵)
245		fiss 9113	. . . 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 583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ (fi‘𝐵))
247	1	ptbasin2 22637	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵)
248	246, 247	sseqtrd 3957	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({𝑋} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (^◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))) ⊆ 𝐵)
249	240, 248	eqssd 3934	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 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 ifcif 4456 𝒫 cpw 4530 {csn 4558 ⟨cop 4564 ∪ cuni 4836 ∩ cint 4876 ∪ ciun 4921 ∩ ciin 4922 ↦ cmpt 5153 × cxp 5578 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 Xcixp 8643 Fincfn 8691 ficfi 9099 Topctop 21950 TopBasesctb 22003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-1o 8267 df-er 8456 df-ixp 8644 df-en 8692 df-dom 8693 df-fin 8695 df-fi 9100 df-top 21951 df-bases 22004

This theorem is referenced by: ptval2 22660 xkoptsub 22713 ptcmplem1 23111 prdsxmslem2 23591

W3C validator