Theorem alexsubALTlem3 22272

Description: Lemma for alexsubALT 22274. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)

Hypothesis

Ref	Expression
alexsubALT.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
alexsubALTlem3	⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ∧ 𝑤 ∈ 𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢)))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑛,𝑠,𝑡,𝑢,𝑤,𝑥,𝑦,𝐽   𝑋,𝑎,𝑏,𝑐,𝑑,𝑛,𝑠,𝑡,𝑢,𝑤,𝑥,𝑦

Proof of Theorem alexsubALTlem3

Dummy variables 𝑓 𝑚 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrex2 3177	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 ↔ ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)
2	1	ralbii 3162	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 ↔ ∀𝑠 ∈ 𝑡 ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)
3		ralnex 3174	. . . . . . . . . 10 ⊢ (∀𝑠 ∈ 𝑡 ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)
4	2, 3	bitr2i 268	. . . . . . . . 9 ⊢ (¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛)
5		elin 4019	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ↔ (𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin))
6		elpwi 4389	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) → 𝑛 ⊆ (𝑢 ∪ {𝑠}))
7	6	adantr 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → 𝑛 ⊆ (𝑢 ∪ {𝑠}))
8		uncom 3980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ∪ {𝑠}) = ({𝑠} ∪ 𝑢)
9	7, 8	syl6sseq 3870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → 𝑛 ⊆ ({𝑠} ∪ 𝑢))
10		ssundif 4276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ⊆ ({𝑠} ∪ 𝑢) ↔ (𝑛 ∖ {𝑠}) ⊆ 𝑢)
11	9, 10	sylib 210	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → (𝑛 ∖ {𝑠}) ⊆ 𝑢)
12		diffi 8482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Fin → (𝑛 ∖ {𝑠}) ∈ Fin)
13	12	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → (𝑛 ∖ {𝑠}) ∈ Fin)
14	11, 13	jca 507	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
15	5, 14	sylbi 209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
16	15	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
17	16	ad2antll 719	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
18		elin 4019	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
19		vex 3401	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑢 ∈ V
20	19	elpw2 5064	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ↔ (𝑛 ∖ {𝑠}) ⊆ 𝑢)
21	20	anbi1i 617	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin) ↔ ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin))
22	18, 21	bitr2i 268	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin) ↔ (𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin))
23	17, 22	sylib 210	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → (𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin))
24		simprrr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 = ∪ 𝑛)
25		eldif 3802	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (𝑛 ∖ {𝑠}) ↔ (𝑥 ∈ 𝑛 ∧ ¬ 𝑥 ∈ {𝑠}))
26	25	simplbi2 496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑛 → (¬ 𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠})))
27		elun 3976	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) ↔ (𝑥 ∈ (𝑛 ∖ {𝑠}) ∨ 𝑥 ∈ {𝑠}))
28		orcom 859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ (𝑥 ∈ (𝑛 ∖ {𝑠}) ∨ 𝑥 ∈ {𝑠}))
29	27, 28	bitr4i 270	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) ↔ (𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠})))
30		df-or 837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ (¬ 𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠})))
31	29, 30	bitr2i 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ 𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}))
32	26, 31	sylib 210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑛 → 𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}))
33	32	ssriv 3825	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ⊆ ((𝑛 ∖ {𝑠}) ∪ {𝑠})
34		uniss 4696	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ⊆ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) → ∪ 𝑛 ⊆ ∪ ((𝑛 ∖ {𝑠}) ∪ {𝑠}))
35	33, 34	mp1i 13	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑛 ⊆ ∪ ((𝑛 ∖ {𝑠}) ∪ {𝑠}))
36		uniun 4694	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) = (∪ (𝑛 ∖ {𝑠}) ∪ ∪ {𝑠})
37		vex 3401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑠 ∈ V
38	37	unisn 4689	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ {𝑠} = 𝑠
39	38	uneq2i 3987	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ (𝑛 ∖ {𝑠}) ∪ ∪ {𝑠}) = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)
40	36, 39	eqtri 2802	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)
41	35, 40	syl6sseq 3870	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑛 ⊆ (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠))
42	24, 41	eqsstrd 3858	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 ⊆ (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠))
43		difss 3960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∖ {𝑠}) ⊆ 𝑛
44	43	unissi 4698	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ (𝑛 ∖ {𝑠}) ⊆ ∪ 𝑛
45		sseq2 3846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 = ∪ 𝑛 → (∪ (𝑛 ∖ {𝑠}) ⊆ 𝑋 ↔ ∪ (𝑛 ∖ {𝑠}) ⊆ ∪ 𝑛))
46	44, 45	mpbiri 250	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ∪ 𝑛 → ∪ (𝑛 ∖ {𝑠}) ⊆ 𝑋)
47	46	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ∪ (𝑛 ∖ {𝑠}) ⊆ 𝑋)
48	47	ad2antll 719	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ (𝑛 ∖ {𝑠}) ⊆ 𝑋)
49		elinel1 4022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ∈ 𝒫 𝑥)
50	49	elpwid 4391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ⊆ 𝑥)
51	50	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) → 𝑡 ⊆ 𝑥)
52	51	ad2antlr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑡 ⊆ 𝑥)
53		simprl 761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ∈ 𝑡)
54	52, 53	sseldd 3822	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ∈ 𝑥)
55		elssuni 4704	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ 𝑥 → 𝑠 ⊆ ∪ 𝑥)
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ⊆ ∪ 𝑥)
57		fibas 21200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (fi‘𝑥) ∈ TopBases
58		unitg 21190	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((fi‘𝑥) ∈ TopBases → ∪ (topGen‘(fi‘𝑥)) = ∪ (fi‘𝑥))
59	57, 58	mp1i 13	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ (topGen‘(fi‘𝑥)) = ∪ (fi‘𝑥))
60		unieq 4681	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ 𝐽 = ∪ (topGen‘(fi‘𝑥)))
61	60	3ad2ant1 1124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → ∪ 𝐽 = ∪ (topGen‘(fi‘𝑥)))
62	61	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝐽 = ∪ (topGen‘(fi‘𝑥)))
63		vex 3401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
64		fiuni 8624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ V → ∪ 𝑥 = ∪ (fi‘𝑥))
65	63, 64	mp1i 13	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 = ∪ (fi‘𝑥))
66	59, 62, 65	3eqtr4rd 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 = ∪ 𝐽)
67		alexsubALT.1	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑋 = ∪ 𝐽
68	66, 67	syl6eqr 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 = 𝑋)
69	56, 68	sseqtrd 3860	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ⊆ 𝑋)
70	48, 69	unssd 4012	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠) ⊆ 𝑋)
71	42, 70	eqssd 3838	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠))
72		unieq 4681	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑛 ∖ {𝑠}) → ∪ 𝑚 = ∪ (𝑛 ∖ {𝑠}))
73	72	uneq1d 3989	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑛 ∖ {𝑠}) → (∪ 𝑚 ∪ 𝑠) = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠))
74	73	rspceeqv 3529	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))
75	23, 71, 74	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))
76	75	expr 450	. . . . . . . . . . . . 13 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)))
77	76	expd 406	. . . . . . . . . . . 12 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) → (𝑋 = ∪ 𝑛 → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))))
78	77	rexlimdv 3212	. . . . . . . . . . 11 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → (∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)))
79	78	ralimdva 3144	. . . . . . . . . 10 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)))
80		elinel2 4023	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ∈ Fin)
81	80	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) → 𝑡 ∈ Fin)
82		unieq 4681	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑓‘𝑠) → ∪ 𝑚 = ∪ (𝑓‘𝑠))
83	82	uneq1d 3989	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑓‘𝑠) → (∪ 𝑚 ∪ 𝑠) = (∪ (𝑓‘𝑠) ∪ 𝑠))
84	83	eqeq2d 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑓‘𝑠) → (𝑋 = (∪ 𝑚 ∪ 𝑠) ↔ 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))
85	84	ac6sfi 8494	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Fin ∧ ∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))
86	85	ex 403	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ Fin → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))))
87	81, 86	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))))
88	87	adantr 474	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))))
89	88	ad2antrl 718	. . . . . . . . . 10 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))))
90		ffvelrn 6623	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin))
91		elin 4019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((𝑓‘𝑠) ∈ 𝒫 𝑢 ∧ (𝑓‘𝑠) ∈ Fin))
92		elpwi 4389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑠) ∈ 𝒫 𝑢 → (𝑓‘𝑠) ⊆ 𝑢)
93	92	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓‘𝑠) ∈ 𝒫 𝑢 ∧ (𝑓‘𝑠) ∈ Fin) → (𝑓‘𝑠) ⊆ 𝑢)
94	91, 93	sylbi 209	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin) → (𝑓‘𝑠) ⊆ 𝑢)
95	90, 94	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ⊆ 𝑢)
96	95	ralrimiva 3148	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢)
97		iunss 4796	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢 ↔ ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢)
98	96, 97	sylibr 226	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢)
99	98	ad2antrl 718	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢)
100		simplrr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ∈ 𝑢)
101	100	snssd 4573	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → {𝑤} ⊆ 𝑢)
102	99, 101	unssd 4012	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢)
103	90	elin2d 4026	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ∈ Fin)
104	103	ralrimiva 3148	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin)
105		iunfi 8544	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ Fin ∧ ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin)
106	81, 104, 105	syl2an 589	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ 𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin)) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin)
107	106	ad4ant14 742	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) ∧ 𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin)) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin)
108	107	ad2ant2lr 738	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin)
109		snfi 8328	. . . . . . . . . . . . . . . 16 ⊢ {𝑤} ∈ Fin
110		unfi 8517	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin ∧ {𝑤} ∈ Fin) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin)
111	108, 109, 110	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin)
112	102, 111	jca 507	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin))
113		elin 4019	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ∧ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin))
114	19	elpw2 5064	. . . . . . . . . . . . . . . 16 ⊢ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢)
115	114	anbi1i 617	. . . . . . . . . . . . . . 15 ⊢ (((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ∧ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) ↔ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin))
116	113, 115	bitr2i 268	. . . . . . . . . . . . . 14 ⊢ (((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) ↔ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin))
117	112, 116	sylib 210	. . . . . . . . . . . . 13 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin))
118		ralnex 3174	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠))
119	118	imbi2i 328	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ (𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)))
120	119	albii 1863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ ∀𝑦(𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)))
121		alinexa 1888	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦(𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ↔ ¬ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)))
122	120, 121	bitr2i 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ↔ ∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)))
123		fveq2 6448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑠 = 𝑧 → (𝑓‘𝑠) = (𝑓‘𝑧))
124	123	unieqd 4683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑠 = 𝑧 → ∪ (𝑓‘𝑠) = ∪ (𝑓‘𝑧))
125		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧)
126	124, 125	uneq12d 3991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑠 = 𝑧 → (∪ (𝑓‘𝑠) ∪ 𝑠) = (∪ (𝑓‘𝑧) ∪ 𝑧))
127	126	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 = 𝑧 → (𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) ↔ 𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧)))
128	127	rspcv 3507	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → 𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧)))
129		eleq2 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 ↔ 𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧)))
130	129	biimpd 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 → 𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧)))
131		elun 3976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) ↔ (𝑣 ∈ ∪ (𝑓‘𝑧) ∨ 𝑣 ∈ 𝑧))
132		eluni 4676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ ∪ (𝑓‘𝑧) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)))
133	132	orbi1i 900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑣 ∈ ∪ (𝑓‘𝑧) ∨ 𝑣 ∈ 𝑧) ↔ (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧))
134		df-or 837	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧) ↔ (¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧))
135		alinexa 1888	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) ↔ ¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)))
136	135	imbi1i 341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧) ↔ (¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧))
137	134, 136	bitr4i 270	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧) ↔ (∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧))
138	131, 133, 137	3bitri 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) ↔ (∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧))
139		eleq2 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 = 𝑤 → (𝑣 ∈ 𝑦 ↔ 𝑣 ∈ 𝑤))
140		eleq1w 2842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑓‘𝑠) ↔ 𝑤 ∈ (𝑓‘𝑠)))
141	140	notbid 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 = 𝑤 → (¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ¬ 𝑤 ∈ (𝑓‘𝑠)))
142	141	ralbidv 3168	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 = 𝑤 → (∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠)))
143	139, 142	imbi12d 336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = 𝑤 → ((𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ (𝑣 ∈ 𝑤 → ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠))))
144	143	spv 2358	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑤 → ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠)))
145	123	eleq2d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑠 = 𝑧 → (𝑤 ∈ (𝑓‘𝑠) ↔ 𝑤 ∈ (𝑓‘𝑧)))
146	145	notbid 310	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑠 = 𝑧 → (¬ 𝑤 ∈ (𝑓‘𝑠) ↔ ¬ 𝑤 ∈ (𝑓‘𝑧)))
147	146	rspcv 3507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠) → ¬ 𝑤 ∈ (𝑓‘𝑧)))
148	144, 147	syl9r 78	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧))))
149	148	alrimdv 1972	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → ∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧))))
150	149	imim1d 82	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑧 ∈ 𝑡 → ((∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))
151	138, 150	syl5bi 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∈ 𝑡 → (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))
152	151	a1dd 50	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∈ 𝑡 → (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))))
153	130, 152	syl9r 78	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝑡 → (𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))))
154	128, 153	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ 𝑋 → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))))
155	154	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = ∩ 𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ 𝑋 → (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))))
156	155	imp31 410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))
157	156	com23 86	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑧 ∈ 𝑡 → 𝑣 ∈ 𝑧)))
158	157	ralrimdv 3150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → ∀𝑧 ∈ 𝑡 𝑣 ∈ 𝑧))
159		vex 3401	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑣 ∈ V
160	159	elint2 4719	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ ∩ 𝑡 ↔ ∀𝑧 ∈ 𝑡 𝑣 ∈ 𝑧)
161	158, 160	syl6ibr 244	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ ∩ 𝑡))
162		eleq2 2848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ∩ 𝑡 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ ∩ 𝑡))
163	162	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ ∩ 𝑡))
164	161, 163	sylibrd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑤))
165	122, 164	syl5bi 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (¬ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑤))
166	165	orrd 852	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤))
167	166	ex 403	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → (∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤)))
168		orc 856	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) → (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))
169	168	anim2i 610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
170	169	eximi 1878	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
171		equid 2059	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑤 = 𝑤
172		vex 3401	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑤 ∈ V
173		equequ1 2072	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑤 → (𝑦 = 𝑤 ↔ 𝑤 = 𝑤))
174	139, 173	anbi12d 624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑤 → ((𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) ↔ (𝑣 ∈ 𝑤 ∧ 𝑤 = 𝑤)))
175	172, 174	spcev 3502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 = 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤))
176	171, 175	mpan2 681	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝑤 → ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤))
177		olc 857	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑤 → (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))
178	177	anim2i 610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) → (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
179	178	eximi 1878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
180	176, 179	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝑤 → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
181	170, 180	jaoi 846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
182		eluni 4676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})))
183		elun 3976	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (𝑦 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑦 ∈ {𝑤}))
184		eliun 4759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ↔ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠))
185		velsn 4414	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤)
186	184, 185	orbi12i 901	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑦 ∈ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))
187	183, 186	bitri 267	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))
188	187	anbi2i 616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) ↔ (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
189	188	exbii 1892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)))
190	182, 189	bitr2i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) ↔ 𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))
191	181, 190	sylib 210	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤) → 𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))
192	167, 191	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 = ∩ 𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})))
193	192	ad5ant25 752	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})))
194	193	ad2ant2l 736	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})))
195	194	ssrdv 3827	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑋 ⊆ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))
196		elun 3976	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (𝑣 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑣 ∈ {𝑤}))
197		eliun 4759	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ↔ ∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠))
198		velsn 4414	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ {𝑤} ↔ 𝑣 = 𝑤)
199	197, 198	orbi12i 901	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑣 ∈ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤))
200	196, 199	bitri 267	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤))
201		nfra1 3123	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)
202		nfv 1957	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠 𝑣 ⊆ 𝑋
203		rsp 3111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑠 ∈ 𝑡 → 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))
204		eqimss2 3877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋)
205		elssuni 4704	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ ∪ (𝑓‘𝑠))
206		ssun3 4001	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ⊆ ∪ (𝑓‘𝑠) → 𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠))
207	205, 206	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠))
208		sstr 3829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠) ∧ (∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋) → 𝑣 ⊆ 𝑋)
209	208	expcom 404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋 → (𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠) → 𝑣 ⊆ 𝑋))
210	204, 207, 209	syl2im 40	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋))
211	203, 210	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑠 ∈ 𝑡 → (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋)))
212	201, 202, 211	rexlimd 3208	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋))
213	212	ad2antll 719	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋))
214		elpwi 4389	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ 𝒫 (fi‘𝑥) → 𝑢 ⊆ (fi‘𝑥))
215	214	ad2antrl 718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → 𝑢 ⊆ (fi‘𝑥))
216	215	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑢 ⊆ (fi‘𝑥))
217	216, 100	sseldd 3822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ∈ (fi‘𝑥))
218		elssuni 4704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ (fi‘𝑥) → 𝑤 ⊆ ∪ (fi‘𝑥))
219	217, 218	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ⊆ ∪ (fi‘𝑥))
220	57, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ (topGen‘(fi‘𝑥)) = ∪ (fi‘𝑥)
221	60, 220	syl6req 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ (fi‘𝑥) = ∪ 𝐽)
222	221, 67	syl6eqr 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ (fi‘𝑥) = 𝑋)
223	222	3ad2ant1 1124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → ∪ (fi‘𝑥) = 𝑋)
224	223	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ (fi‘𝑥) = 𝑋)
225	219, 224	sseqtrd 3860	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ⊆ 𝑋)
226		sseq1 3845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑣 ⊆ 𝑋 ↔ 𝑤 ⊆ 𝑋))
227	225, 226	syl5ibrcom 239	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 = 𝑤 → 𝑣 ⊆ 𝑋))
228	213, 227	jaod 848	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ((∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤) → 𝑣 ⊆ 𝑋))
229	200, 228	syl5bi 234	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) → 𝑣 ⊆ 𝑋))
230	229	ralrimiv 3147	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∀𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})𝑣 ⊆ 𝑋)
231		unissb 4706	. . . . . . . . . . . . . . 15 ⊢ (∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑋 ↔ ∀𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})𝑣 ⊆ 𝑋)
232	230, 231	sylibr 226	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑋)
233	195, 232	eqssd 3838	. . . . . . . . . . . . 13 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑋 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))
234		unieq 4681	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) → ∪ 𝑏 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))
235	234	rspceeqv 3529	. . . . . . . . . . . . 13 ⊢ (((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)
236	117, 233, 235	syl2anc 579	. . . . . . . . . . . 12 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)
237	236	ex 403	. . . . . . . . . . 11 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏))
238	237	exlimdv 1976	. . . . . . . . . 10 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏))
239	79, 89, 238	3syld 60	. . . . . . . . 9 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏))
240	4, 239	syl5bi 234	. . . . . . . 8 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏))
241		dfrex2 3177	. . . . . . . 8 ⊢ (∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏 ↔ ¬ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)
242	240, 241	syl6ib 243	. . . . . . 7 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 → ¬ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))
243	242	con4d 115	. . . . . 6 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛))
244	243	exp32 413	. . . . 5 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → (𝑤 ∈ 𝑢 → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛))))
245	244	com24 95	. . . 4 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛))))
246	245	exp32 413	. . 3 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → (𝑢 ∈ 𝒫 (fi‘𝑥) → (𝑎 ⊆ 𝑢 → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛))))))
247	246	imp45 422	. 2 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)))
248	247	imp31 410	1 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ∧ 𝑤 ∈ 𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢)))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792  𝒫 cpw 4379  {csn 4398  ∪ cuni 4673  ∩ cint 4712  ∪ ciun 4755  ⟶wf 6133  ‘cfv 6137  Fincfn 8243  ficfi 8606  topGenctg 16495  TopBasesctb 21168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-fin 8247  df-fi 8607  df-topgen 16501  df-bases 21169

This theorem is referenced by:  alexsubALTlem4  22273