Theorem cmpfi 22016

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 4548	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐽 → 𝑦 ⊆ 𝐽)
2		0ss 4350	. . . . . . . . . . 11 ⊢ ∅ ⊆ 𝑦
3		0fin 8746	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
4		elfpw 8826	. . . . . . . . . . 11 ⊢ (∅ ∈ (𝒫 𝑦 ∩ Fin) ↔ (∅ ⊆ 𝑦 ∧ ∅ ∈ Fin))
5	2, 3, 4	mpbir2an 709	. . . . . . . . . 10 ⊢ ∅ ∈ (𝒫 𝑦 ∩ Fin)
6		simprr 771	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝐽 = ∪ 𝑦)
7		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → 𝑦 = ∅)
8	7	unieqd 4852	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝑦 = ∪ ∅)
9	6, 8	eqtrd 2856	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∪ 𝐽 = ∪ ∅)
10		unieq 4849	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
11	10	rspceeqv 3638	. . . . . . . . . 10 ⊢ ((∅ ∈ (𝒫 𝑦 ∩ Fin) ∧ ∪ 𝐽 = ∪ ∅) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
12	5, 9, 11	sylancr 589	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 = ∅ ∧ ∪ 𝐽 = ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
13	12	expr 459	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
14		vn0 4304	. . . . . . . . . 10 ⊢ V ≠ ∅
15		iineq1 4936	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟))
16	15	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟))
17		0iin 4987	. . . . . . . . . . . . 13 ⊢ ∩ 𝑟 ∈ ∅ (∪ 𝐽 ∖ 𝑟) = V
18	16, 17	syl6eq 2872	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = V)
19	18	eqeq1d 2823	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ V = ∅))
20	19	necon3bbid 3053	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (¬ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ V ≠ ∅))
21	14, 20	mpbiri 260	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ¬ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅)
22	21	pm2.21d 121	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
23	13, 22	2thd 267	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 = ∅) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
24		uniss 4846	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝐽 → ∪ 𝑦 ⊆ ∪ 𝐽)
25	24	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∪ 𝑦 ⊆ ∪ 𝐽)
26		eqss 3982	. . . . . . . . . . . 12 ⊢ (∪ 𝑦 = ∪ 𝐽 ↔ (∪ 𝑦 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ ∪ 𝑦))
27	26	baib 538	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ⊆ ∪ 𝐽 → (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑦))
28	25, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑦))
29		eqcom 2828	. . . . . . . . . 10 ⊢ (∪ 𝑦 = ∪ 𝐽 ↔ ∪ 𝐽 = ∪ 𝑦)
30		ssdif0 4323	. . . . . . . . . 10 ⊢ (∪ 𝐽 ⊆ ∪ 𝑦 ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅)
31	28, 29, 30	3bitr3g 315	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∪ 𝑦 ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅))
32		iindif2 4999	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ ∅ → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟))
33	32	adantl 484	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟))
34		uniiun 4982	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 = ∪ 𝑟 ∈ 𝑦 𝑟
35	34	difeq2i 4096	. . . . . . . . . . 11 ⊢ (∪ 𝐽 ∖ ∪ 𝑦) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑦 𝑟)
36	33, 35	syl6eqr 2874	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑦))
37	36	eqeq1d 2823	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ (∪ 𝐽 ∖ ∪ 𝑦) = ∅))
38	31, 37	bitr4d 284	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∪ 𝑦 ↔ ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅))
39		imassrn 5940	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ⊆ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
40		df-ima 5568	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦)
41		resmpt 5905	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐽 → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
42	41	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
43	42	rneqd 5808	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ran ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
44	40, 43	syl5eq 2868	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
45	44	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
46	39, 45	sseqtrrid 4020	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
47		funmpt 6393	. . . . . . . . . . . 12 ⊢ Fun (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
48		elinel2 4173	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) → 𝑧 ∈ Fin)
49	48	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → 𝑧 ∈ Fin)
50		imafi 8817	. . . . . . . . . . . 12 ⊢ ((Fun (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ∧ 𝑧 ∈ Fin) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ Fin)
51	47, 49, 50	sylancr 589	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ Fin)
52		elfpw 8826	. . . . . . . . . . 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))
53	46, 51, 52	sylanbrc 585	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin))
54		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
55	54	topopn 21514	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽)
56		difexg 5231	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝐽 ∈ 𝐽 → (∪ 𝐽 ∖ 𝑟) ∈ V)
57	55, 56	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ 𝑟) ∈ V)
58	57	ralrimivw 3183	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → ∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V)
59		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟))
60	59	fnmpt 6488	. . . . . . . . . . . . . 14 ⊢ (∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
61	58, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
62	61	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) Fn 𝑦)
63		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin))
64		elfpw 8826	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin) ↔ (𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
65	63, 64	sylib 220	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → (𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
66	65	simpld 497	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
67	44	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
68	66, 67	sseqtrd 4007	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
69	65	simprd 498	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ Fin)
70		fipreima 8830	. . . . . . . . . . . 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)
71	62, 68, 69, 70	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤)
72		eqcom 2828	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤 ↔ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
73	72	rexbii 3247	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
74	71, 73	sylib 220	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
75		simpr 487	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
76	75	inteqd 4881	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → ∩ 𝑤 = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
77	76	eqeq2d 2832	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) → (∅ = ∩ 𝑤 ↔ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
78	53, 74, 77	rexxfrd 5310	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)∅ = ∩ 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
79		0ex 5211	. . . . . . . . . 10 ⊢ ∅ ∈ V
80		imassrn 5940	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
81		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
82	54, 81	opncldf1 21692	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡(𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (𝑣 ∈ (Clsd‘𝐽) ↦ (∪ 𝐽 ∖ 𝑣))))
83	82	simpld 497	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽))
84		f1ofo 6622	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–1-1-onto→(Clsd‘𝐽) → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽))
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽))
86		forn 6593	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽) → ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (Clsd‘𝐽))
87	85, 86	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ran (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) = (Clsd‘𝐽))
88	80, 87	sseqtrid 4019	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
89		fvex 6683	. . . . . . . . . . . . 13 ⊢ (Clsd‘𝐽) ∈ V
90	89	elpw2 5248	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽) ↔ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
91	88, 90	sylibr 236	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
92	91	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
93		elfi 8877	. . . . . . . . . 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))
94	79, 92, 93	sylancr 589	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∩ Fin)∅ = ∩ 𝑤))
95		inundif 4427	. . . . . . . . . . . . . 14 ⊢ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) = (𝒫 𝑦 ∩ Fin)
96	95	rexeqi 3414	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)
97		rexun 4166	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))∪ 𝐽 = ∪ 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
98	96, 97	bitr3i 279	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
99		elinel2 4173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 ∈ {∅})
100		elsni 4584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
101	99, 100	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
102	101	unieqd 4852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ∪ 𝑧 = ∪ ∅)
103		uni0 4866	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ ∅ = ∅
104	102, 103	syl6eq 2872	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ∪ 𝑧 = ∅)
105	104	eqeq2d 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → (∪ 𝐽 = ∪ 𝑧 ↔ ∪ 𝐽 = ∅))
106	105	biimpd 231	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → (∪ 𝐽 = ∪ 𝑧 → ∪ 𝐽 = ∅))
107	106	rexlimiv 3280	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 → ∪ 𝐽 = ∅)
108		ssidd 3990	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ⊆ 𝑦)
109		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 = ∅)
110		0ss 4350	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∅ ⊆ ∪ 𝑦
111	109, 110	eqsstrdi 4021	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 ⊆ ∪ 𝑦)
112	24	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 ⊆ ∪ 𝐽)
113	111, 112	eqssd 3984	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝐽 = ∪ 𝑦)
114	113, 109	eqtr3d 2858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 = ∅)
115	114, 3	eqeltrdi 2921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∪ 𝑦 ∈ Fin)
116		pwfi 8819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ 𝑦 ∈ Fin ↔ 𝒫 ∪ 𝑦 ∈ Fin)
117	115, 116	sylib 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝒫 ∪ 𝑦 ∈ Fin)
118		pwuni 4875	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ⊆ 𝒫 ∪ 𝑦
119		ssfi 8738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝒫 ∪ 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝒫 ∪ 𝑦) → 𝑦 ∈ Fin)
120	117, 118, 119	sylancl 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ Fin)
121		elfpw 8826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ⊆ 𝑦 ∧ 𝑦 ∈ Fin))
122	108, 120, 121	sylanbrc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ (𝒫 𝑦 ∩ Fin))
123		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ≠ ∅)
124		eldifsn 4719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑦 ≠ ∅))
125	122, 123, 124	sylanbrc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → 𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))
126		unieq 4849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → ∪ 𝑧 = ∪ 𝑦)
127	126	rspceeqv 3638	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∪ 𝐽 = ∪ 𝑦) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧)
128	125, 113, 127	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ (𝑦 ≠ ∅ ∧ ∪ 𝐽 = ∅)) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧)
129	128	expr 459	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∪ 𝐽 = ∅ → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
130	107, 129	syl5 34	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
131		idd 24	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
132	130, 131	jaod 855	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
133		olc 864	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
134	132, 133	impbid1 227	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅})∪ 𝐽 = ∪ 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
135	98, 134	syl5bb 285	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧))
136		eldifi 4103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
137	136	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
138		elfpw 8826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑧 ⊆ 𝑦 ∧ 𝑧 ∈ Fin))
139	137, 138	sylib 220	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (𝑧 ⊆ 𝑦 ∧ 𝑧 ∈ Fin))
140	139	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ⊆ 𝑦)
141		simpllr 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑦 ⊆ 𝐽)
142	140, 141	sstrd 3977	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ⊆ 𝐽)
143	142	unissd 4848	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∪ 𝑧 ⊆ ∪ 𝐽)
144		eqss 3982	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑧 = ∪ 𝐽 ↔ (∪ 𝑧 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ⊆ ∪ 𝑧))
145	144	baib 538	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑧 ⊆ ∪ 𝐽 → (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑧))
146	143, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 ⊆ ∪ 𝑧))
147		eqcom 2828	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 = ∪ 𝐽 ↔ ∪ 𝐽 = ∪ 𝑧)
148		ssdif0 4323	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ⊆ ∪ 𝑧 ↔ (∪ 𝐽 ∖ ∪ 𝑧) = ∅)
149		eqcom 2828	. . . . . . . . . . . . . . 15 ⊢ ((∪ 𝐽 ∖ ∪ 𝑧) = ∅ ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧))
150	148, 149	bitri 277	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐽 ⊆ ∪ 𝑧 ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧))
151	146, 147, 150	3bitr3g 315	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝐽 = ∪ 𝑧 ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧)))
152		df-ima 5568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ran ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧)
153	140	resmptd 5908	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧) = (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
154	153	rneqd 5808	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ran ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) ↾ 𝑧) = ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
155	152, 154	syl5eq 2868	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
156	155	inteqd 4881	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
157	57	ralrimivw 3183	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ Top → ∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V)
158	157	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V)
159		dfiin3g 5836	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) ∈ V → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
160	158, 159	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑧 ↦ (∪ 𝐽 ∖ 𝑟)))
161		eldifsni 4722	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
162	161	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ≠ ∅)
163		iindif2 4999	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
164	162, 163	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ 𝑟 ∈ 𝑧 (∪ 𝐽 ∖ 𝑟) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
165	156, 160, 164	3eqtr2d 2862	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟))
166		uniiun 4982	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑟 ∈ 𝑧 𝑟
167	166	difeq2i 4096	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝐽 ∖ ∪ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑟 ∈ 𝑧 𝑟)
168	165, 167	syl6eqr 2874	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = (∪ 𝐽 ∖ ∪ 𝑧))
169	168	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ↔ ∅ = (∪ 𝐽 ∖ ∪ 𝑧)))
170	151, 169	bitr4d 284	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∪ 𝐽 = ∪ 𝑧 ↔ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
171	170	rexbidva 3296	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
172	135, 171	bitrd 281	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
173		imaeq2 5925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ∅ → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ ∅))
174		ima0 5945	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ ∅) = ∅
175	173, 174	syl6eq 2872	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ∅ → ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∅)
176	175	inteqd 4881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = ∩ ∅)
177		int0 4890	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ ∅ = V
178	176, 177	syl6eq 2872	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) = V)
179	178	neeq1d 3075	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ∅ → (∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ≠ ∅ ↔ V ≠ ∅))
180	14, 179	mpbiri 260	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ∅ → ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ≠ ∅)
181	180	necomd 3071	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → ∅ ≠ ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
182	181	necon2i 3050	. . . . . . . . . . . . 13 ⊢ (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) → 𝑧 ≠ ∅)
183		eldifsn 4719	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑧 ≠ ∅))
184	183	rbaibr 540	. . . . . . . . . . . . 13 ⊢ (𝑧 ≠ ∅ → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
185	182, 184	syl 17	. . . . . . . . . . . 12 ⊢ (∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
186	185	pm5.32ri 578	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)) ↔ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
187	186	rexbii2 3245	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧))
188	172, 187	syl6bbr 291	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ∩ ((𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑧)))
189	78, 94, 188	3bitr4rd 314	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧 ↔ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
190	38, 189	imbi12d 347	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) ∧ 𝑦 ≠ ∅) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
191	23, 190	pm2.61dane 3104	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
192	58	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V)
193		dfiin3g 5836	. . . . . . . . . . 11 ⊢ (∀𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) ∈ V → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
194	192, 193	syl 17	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
195	44	inteqd 4881	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∩ ran (𝑟 ∈ 𝑦 ↦ (∪ 𝐽 ∖ 𝑟)))
196	194, 195	eqtr4d 2859	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
197	196	eqeq1d 2823	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∅))
198		nne 3020	. . . . . . . 8 ⊢ (¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) = ∅)
199	197, 198	syl6bbr 291	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → (∩ 𝑟 ∈ 𝑦 (∪ 𝐽 ∖ 𝑟) = ∅ ↔ ¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
200	199	imbi1d 344	. . . . . 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)))))
201	191, 200	bitrd 281	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))))
202		con1b 361	. . . . 5 ⊢ ((¬ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
203	201, 202	syl6bb 289	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ⊆ 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
204	1, 203	sylan2 594	. . 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) ≠ ∅)))
205	204	ralbidva 3196	. 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) ≠ ∅)))
206	54	iscmp 21996	. . 3 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
207	206	baib 538	. 2 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
208	91	adantr 483	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
209		simpl 485	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝐽 ∈ Top)
210		funmpt 6393	. . . . . 6 ⊢ Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟))
211	210	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)))
212		elpwi 4548	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
213		foima 6595	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)):𝐽–onto→(Clsd‘𝐽) → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) = (Clsd‘𝐽))
214	85, 213	syl 17	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) = (Clsd‘𝐽))
215	214	sseq2d 3999	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽) ↔ 𝑥 ⊆ (Clsd‘𝐽)))
216	212, 215	syl5ibr 248	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽)))
217	216	imp 409	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽))
218		ssimaexg 6749	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ Fun (𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) ∧ 𝑥 ⊆ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝐽)) → ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
219	209, 211, 217, 218	syl3anc 1367	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
220		df-rex 3144	. . . . 5 ⊢ (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
221		velpw 4544	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐽 ↔ 𝑦 ⊆ 𝐽)
222	221	anbi1i 625	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ (𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
223	222	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
224	220, 223	bitri 277	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ⊆ 𝐽 ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
225	219, 224	sylibr 236	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
226		simpr 487	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
227	226	fveq2d 6674	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (fi‘𝑥) = (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)))
228	227	eleq2d 2898	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
229	228	notbid 320	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))))
230	226	inteqd 4881	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ 𝑥 = ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦))
231	230	neeq1d 3075	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → (∩ 𝑥 ≠ ∅ ↔ ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅))
232	229, 231	imbi12d 347	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ((¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
233	208, 225, 232	ralxfrd 5309	. 2 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦)) → ∩ ((𝑟 ∈ 𝐽 ↦ (∪ 𝐽 ∖ 𝑟)) “ 𝑦) ≠ ∅)))
234	205, 207, 233	3bitr4d 313	1 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ∪ cuni 4838  ∩ cint 4876  ∪ ciun 4919  ∩ ciin 4920   ↦ cmpt 5146  ^◡ccnv 5554  ran crn 5556   ↾ cres 5557   “ cima 5558  Fun wfun 6349   Fn wfn 6350  –onto→wfo 6353  –1-1-onto→wf1o 6354  ‘cfv 6355  Fincfn 8509  ficfi 8874  Topctop 21501  Clsdccld 21624  Compccmp 21994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fi 8875  df-top 21502  df-cld 21627  df-cmp 21995

This theorem is referenced by:  cmpfii  22017  fclscmp  22638  heibor1lem  35102