Theorem cmpfi 21211

Description: If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Assertion

Ref	Expression
cmpfi	⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))

Distinct variable group:   𝑥,𝐽

Proof of Theorem cmpfi

Dummy variables 𝑣 𝑟 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4168	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐽 → 𝑦 ⊆ 𝐽)
2		0ss 3972	. . . . . . . . . . 11 ⊢ ∅ ⊆ 𝑦
3		0fin 8188	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
4		elfpw 8268	. . . . . . . . . . 11 ⊢ (∅ ∈ (𝒫 𝑦 ∩ Fin) ↔ (∅ ⊆ 𝑦 ∧ ∅ ∈ Fin))
5	2, 3, 4	mpbir2an 955	. . . . . . . . . 10 ⊢ ∅ ∈ (𝒫 𝑦 ∩ Fin)
6		simprr 796	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝐽 = ∪ 𝑦)
7		simprl 794	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → 𝑦 = ∅)
8	7	unieqd 4446	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝑦 = ∪ ∅)
9	6, 8	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝐽 = ∪ ∅)
10		unieq 4444	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
11	10	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∪ ∅))
12	11	rspcev 3309	. . . . . . . . . 10 ⊢ ((∅ ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ ∅) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
13	5, 9, 12	sylancr 695	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
14	13	expr 643	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
15		vn0 3924	. . . . . . . . . 10 ⊢ V ≠ ∅
16		iineq1 4535	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟))
17	16	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟))
18		0iin 4578	. . . . . . . . . . . . 13 ⊢ ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟) = V
19	17, 18	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = V)
20	19	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ V = ∅))
21	20	necon3bbid 2831	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (¬ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ V ≠ ∅))
22	15, 21	mpbiri 248	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ¬ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅)
23	22	pm2.21d 118	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
24	14, 23	2thd 255	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
25		uniss 4458	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝐽 → ∪ 𝑦 ⊆ ∪ 𝐽)
26	25	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∪ 𝑦 ⊆ ∪ 𝐽)
27		eqss 3618	. . . . . . . . . . . 12 ⊢ (∪ 𝑦 = ∪ 𝐽 ↔ (∪ 𝑦 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ ∪ 𝑦))
28	27	baib 944	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ⊆ ∪ 𝐽 → (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑦))
29	26, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑦))
30		eqcom 2629	. . . . . . . . . 10 ⊢ (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 = ∪ 𝑦)
31		ssdif0 3942	. . . . . . . . . 10 ⊢ (∪ 𝐽 ⊆ ∪ 𝑦 ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅)
32	29, 30, 31	3bitr3g 302	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∪ 𝑦 ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅))
33		iindif2 4589	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ ∅ → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟))
34	33	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟))
35		uniiun 4573	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 = ∪ 𝑟 ∈ 𝑦 𝑟
36	35	difeq2i 3725	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ ∪ 𝑦) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟)
37	34, 36	syl6eqr 2674	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑦))
38	37	eqeq1d 2624	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅))
39	32, 38	bitr4d 271	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∪ 𝑦 ↔ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅))
40		imassrn 5477	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ⊆ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
41		df-ima 5127	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦)
42		resmpt 5449	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐽 → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
43	42	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
44	43	rneqd 5353	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ran ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
45	41, 44	syl5eq 2668	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
46	45	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
47	40, 46	syl5sseqr 3654	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
48		funmpt 5926	. . . . . . . . . . . 12 ⊢ Fun (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
49		elfpw 8268	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑧 ⊆ 𝑦 ∧ 𝑧 ∈ Fin))
50	49	simprbi 480	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) → 𝑧 ∈ Fin)
51	50	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → 𝑧 ∈ Fin)
52		imafi 8259	. . . . . . . . . . . 12 ⊢ ((Fun (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ∧ 𝑧 ∈ Fin) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ Fin)
53	48, 51, 52	sylancr 695	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ Fin)
54		elfpw 8268	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin) ↔ (((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∧ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ Fin))
55	47, 53, 54	sylanbrc 698	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin))
56		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
57	56	topopn 20711	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
58		difexg 4808	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝐽 ∈ 𝐽 → (∪ 𝐽 ∖ 𝑟) ∈ V)
59	57, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ 𝑟) ∈ V)
60	59	ralrimivw 2967	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → ∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V)
61		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
62	61	fnmpt 6020	. . . . . . . . . . . . . 14 ⊢ (∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
63	60, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
64	63	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
65		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin))
66		elfpw 8268	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin) ↔ (𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
67	65, 66	sylib 208	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → (𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
68	67	simpld 475	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
69	45	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
70	68, 69	sseqtrd 3641	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
71	67	simprd 479	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ Fin)
72		fipreima 8272	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦 ∧ 𝑤 ⊆ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ∧ 𝑤 ∈ Fin) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤)
73	64, 70, 71, 72	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤)
74		eqcom 2629	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤 ↔ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
75	74	rexbii 3041	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
76	73, 75	sylib 208	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
77		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
78	77	inteqd 4480	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → ∩ 𝑤 = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
79	78	eqeq2d 2632	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → (∅ = ∩ 𝑤 ↔ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
80	55, 76, 79	rexxfrd 4881	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)∅ = ∩ 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
81		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
82		imassrn 5477	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
83		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
84	56, 83	opncldf1 20888	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡(𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑣 ∈ (Clsd‘𝐽) ↦ (∪ 𝐽 ∖ 𝑣))))
85	84	simpld 475	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽))
86		f1ofo 6144	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽) → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽))
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽))
88		forn 6118	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽) → ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (Clsd‘𝐽))
89	87, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (Clsd‘𝐽))
90	82, 89	syl5sseq 3653	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
91		fvex 6201	. . . . . . . . . . . . 13 ⊢ (Clsd‘𝐽) ∈ V
92	91	elpw2 4828	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽) ↔ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
93	90, 92	sylibr 224	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
94	93	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
95		elfi 8319	. . . . . . . . . 10 ⊢ ((∅ ∈ V ∧ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽)) → (∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)∅ = ∩ 𝑤))
96	81, 94, 95	sylancr 695	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)∅ = ∩ 𝑤))
97		inundif 4046	. . . . . . . . . . . . . 14 ⊢ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) = (𝒫 𝑦 ∩ Fin)
98	97	rexeqi 3143	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
99		rexun 3793	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))∪ 𝐽 = ∪ 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
100	98, 99	bitr3i 266	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
101		inss2 3834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) ⊆ {∅}
102	101	sseli 3599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 ∈ {∅})
103		elsni 4194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
105	104	unieqd 4446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ∪ 𝑧 = ∪ ∅)
106		uni0 4465	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ ∅ = ∅
107	105, 106	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ∪ 𝑧 = ∅)
108	107	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∅))
109	108	biimpd 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → (∪ 𝐽 = ∪ 𝑧 → ∪ 𝐽 = ∅))
110	109	rexlimiv 3027	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 → ∪ 𝐽 = ∅)
111		ssid 3624	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ⊆ 𝑦
112	111	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ⊆ 𝑦)
113		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 = ∅)
114		0ss 3972	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∅ ⊆ ∪ 𝑦
115	113, 114	syl6eqss 3655	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 ⊆ ∪ 𝑦)
116	25	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 ⊆ ∪ 𝐽)
117	115, 116	eqssd 3620	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 = ∪ 𝑦)
118	117, 113	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 = ∅)
119	118, 3	syl6eqel 2709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 ∈ Fin)
120		pwfi 8261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝑦 ∈ Fin ↔ 𝒫 ∪ 𝑦 ∈ Fin)
121	119, 120	sylib 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝒫 ∪ 𝑦 ∈ Fin)
122		pwuni 4474	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ⊆ 𝒫 ∪ 𝑦
123		ssfi 8180	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝒫 ∪ 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝒫 ∪ 𝑦) → 𝑦 ∈ Fin)
124	121, 122, 123	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ Fin)
125		elfpw 8268	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
126	112, 124, 125	sylanbrc 698	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ (𝒫 𝑦 ∩ Fin))
127		simprl 794	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ≠ ∅)
128		eldifsn 4317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑦 ≠ ∅))
129	126, 127, 128	sylanbrc 698	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))
130		unieq 4444	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
131	130	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∪ 𝑦))
132	131	rspcev 3309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧)
133	129, 117, 132	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧)
134	133	expr 643	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∅ → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
135	110, 134	syl5 34	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
136		idd 24	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
137	135, 136	jaod 395	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
138		olc 399	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
139	137, 138	impbid1 215	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
140	100, 139	syl5bb 272	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
141		eldifi 3732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
142	141	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
143	142, 49	sylib 208	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (𝑧 ⊆ 𝑦 ∧ 𝑧 ∈ Fin))
144	143	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ⊆ 𝑦)
145		simpllr 799	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑦 ⊆ 𝐽)
146	144, 145	sstrd 3613	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ⊆ 𝐽)
147	146	unissd 4462	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∪ 𝑧 ⊆ ∪ 𝐽)
148		eqss 3618	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑧 = ∪ 𝐽 ↔ (∪ 𝑧 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ ∪ 𝑧))
149	148	baib 944	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑧 ⊆ ∪ 𝐽 → (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑧))
150	147, 149	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑧))
151		eqcom 2629	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 = ∪ 𝑧)
152		ssdif0 3942	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ⊆ ∪ 𝑧 ↔ (∪ 𝐽 ∖ ∪ 𝑧) = ∅)
153		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝐽 ∖ ∪ 𝑧) = ∅ ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧))
154	152, 153	bitri 264	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐽 ⊆ ∪ 𝑧 ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧))
155	150, 151, 154	3bitr3g 302	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝐽 = ∪ 𝑧 ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧)))
156		df-ima 5127	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ran ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧)
157	144	resmptd 5452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧) = (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
158	157	rneqd 5353	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ran ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧) = ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
159	156, 158	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
160	159	inteqd 4480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
161	59	ralrimivw 2967	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ Top → ∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V)
162	161	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V)
163		dfiin3g 5379	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
164	162, 163	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
165		eldifsn 4317	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑧 ≠ ∅))
166	165	simprbi 480	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
167	166	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ≠ ∅)
168		iindif2 4589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
169	167, 168	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
170	160, 164, 169	3eqtr2d 2662	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
171		uniiun 4573	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑟 ∈ 𝑧 𝑟
172	171	difeq2i 3725	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∖ ∪ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟)
173	170, 172	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑧))
174	173	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧)))
175	155, 174	bitr4d 271	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝐽 = ∪ 𝑧 ↔ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
176	175	rexbidva 3049	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
177	140, 176	bitrd 268	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
178		imaeq2 5462	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ∅ → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ ∅))
179		ima0 5481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ ∅) = ∅
180	178, 179	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∅ → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∅)
181	180	inteqd 4480	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∩ ∅)
182		int0 4490	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ ∅ = V
183	181, 182	syl6eq 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = V)
184	183	neeq1d 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ∅ → (∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ≠ ∅ ↔ V ≠ ∅))
185	15, 184	mpbiri 248	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ≠ ∅)
186	185	necomd 2849	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → ∅ ≠ ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
187	186	necon2i 2828	. . . . . . . . . . . . 13 ⊢ (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) → 𝑧 ≠ ∅)
188	165	rbaibr 946	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
189	187, 188	syl 17	. . . . . . . . . . . 12 ⊢ (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
190	189	pm5.32ri 670	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) ↔ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
191	190	rexbii2 3039	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
192	177, 191	syl6bbr 278	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
193	80, 96, 192	3bitr4rd 301	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
194	39, 193	imbi12d 334	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
195	24, 194	pm2.61dane 2881	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
196	60	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V)
197		dfiin3g 5379	. . . . . . . . . . 11 ⊢ (∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
198	196, 197	syl 17	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
199	45	inteqd 4480	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
200	198, 199	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
201	200	eqeq1d 2624	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∅))
202		nne 2798	. . . . . . . 8 ⊢ (¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∅)
203	201, 202	syl6bbr 278	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ ¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
204	203	imbi1d 331	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))) ↔ (¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
205	195, 204	bitrd 268	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
206		con1b 348	. . . . 5 ⊢ ((¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
207	205, 206	syl6bb 276	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
208	1, 207	sylan2 491	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
209	208	ralbidva 2985	. 2 ⊢ (𝐽 ∈ Top → (∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
210	56	iscmp 21191	. . 3 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
211	210	baib 944	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
212	93	adantr 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
213		simpl 473	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝐽 ∈ Top)
214		funmpt 5926	. . . . . 6 ⊢ Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
215	214	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)))
216		elpwi 4168	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
217		foima 6120	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) = (Clsd‘𝐽))
218	87, 217	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) = (Clsd‘𝐽))
219	218	sseq2d 3633	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) ↔ 𝑥 ⊆ (Clsd‘𝐽)))
220	216, 219	syl5ibr 236	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽)))
221	220	imp 445	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽))
222		ssimaexg 6264	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ∧ 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽)) → ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
223	213, 215, 221, 222	syl3anc 1326	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
224		df-rex 2918	. . . . 5 ⊢ (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
225		selpw 4165	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽)
226	225	anbi1i 731	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ (𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
227	226	exbii 1774	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
228	224, 227	bitri 264	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
229	223, 228	sylibr 224	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
230		simpr 477	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
231	230	fveq2d 6195	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (fi‘𝑥) = (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
232	231	eleq2d 2687	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
233	232	notbid 308	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
234	230	inteqd 4480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ 𝑥 = ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
235	234	neeq1d 2853	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (∩ 𝑥 ≠ ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
236	233, 235	imbi12d 334	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ((¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
237	212, 229, 236	ralxfrd 4879	. 2 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
238	209, 211, 237	3bitr4d 300	1 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∩ cint 4475  ∪ ciun 4520  ∩ ciin 4521   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882   Fn wfn 5883  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  Fincfn 7955  ficfi 8316  Topctop 20698  Clsdccld 20820  Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-top 20699  df-cld 20823  df-cmp 21190

This theorem is referenced by:  cmpfii  21212  fclscmp  21834  heibor1lem  33608