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Theorem fiuncmp 23427

Description: A finite union of compact sets is compact. (Contributed by Mario Carneiro, 19-Mar-2015.)

**Hypothesis**
Ref	Expression
fiuncmp.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
fiuncmp	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐽 Allowed substitution hints: 𝐵(𝑥) 𝑋(𝑥)

Proof of Theorem fiuncmp Dummy variables 𝑡 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 4017	. 2 ⊢ 𝐴 ⊆ 𝐴
2		simp2 1136	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → 𝐴 ∈ Fin)
3		sseq1 4020	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		iuneq1 5012	. . . . . . . . 9 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
5		0iun 5067	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
6	4, 5	eqtrdi 2790	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∅)
7	6	oveq2d 7446	. . . . . . 7 ⊢ (𝑡 = ∅ → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∅))
8	7	eleq1d 2823	. . . . . 6 ⊢ (𝑡 = ∅ → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∅) ∈ Comp))
9	3, 8	imbi12d 344	. . . . 5 ⊢ (𝑡 = ∅ → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (∅ ⊆ 𝐴 → (𝐽 ↾_t ∅) ∈ Comp)))
10	9	imbi2d 340	. . . 4 ⊢ (𝑡 = ∅ → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (∅ ⊆ 𝐴 → (𝐽 ↾_t ∅) ∈ Comp))))
11		sseq1 4020	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12		iuneq1 5012	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
13	12	oveq2d 7446	. . . . . . 7 ⊢ (𝑡 = 𝑦 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵))
14	13	eleq1d 2823	. . . . . 6 ⊢ (𝑡 = 𝑦 → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp))
15	11, 14	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝑦 → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)))
16	15	imbi2d 340	. . . 4 ⊢ (𝑡 = 𝑦 → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp))))
17		sseq1 4020	. . . . . 6 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (𝑡 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
18		iuneq1 5012	. . . . . . . 8 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
19	18	oveq2d 7446	. . . . . . 7 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵))
20	19	eleq1d 2823	. . . . . 6 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))
21	17, 20	imbi12d 344	. . . . 5 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
22	21	imbi2d 340	. . . 4 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))))
23		sseq1 4020	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
24		iuneq1 5012	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
25	24	oveq2d 7446	. . . . . . 7 ⊢ (𝑡 = 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵))
26	25	eleq1d 2823	. . . . . 6 ⊢ (𝑡 = 𝐴 → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp))
27	23, 26	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝐴 → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)))
28	27	imbi2d 340	. . . 4 ⊢ (𝑡 = 𝐴 → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp)) ↔ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp))))
29		rest0 23192	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) = {∅})
30		0cmp 23417	. . . . . . 7 ⊢ {∅} ∈ Comp
31	29, 30	eqeltrdi 2846	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) ∈ Comp)
32	31	3ad2ant1 1132	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐽 ↾_t ∅) ∈ Comp)
33	32	a1d 25	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (∅ ⊆ 𝐴 → (𝐽 ↾_t ∅) ∈ Comp))
34		ssun1 4187	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		id 22	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
36	34, 35	sstrid 4006	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑦 ⊆ 𝐴)
37	36	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp))
38		simpl1 1190	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → 𝐽 ∈ Top)
39		iunxun 5098	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
40		simprr 773	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)
41		cmptop 23418	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top)
42		restrcl 23180	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top → (𝐽 ∈ V ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V))
43	42	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V)
44	40, 41, 43	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V)
45		nfcv 2902	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝐵
46		nfcsb1v 3932	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑡 / 𝑥⦌𝐵
47		csbeq1a 3921	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → 𝐵 = ⦋𝑡 / 𝑥⦌𝐵)
48	45, 46, 47	cbviun 5040	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵
49		vex 3481	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
50		csbeq1 3910	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
51	49, 50	iunxsn 5095	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
52	48, 51	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
53	50	oveq2d 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑧 → (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) = (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵))
54	53	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ((𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp ↔ (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp))
55		simpl3 1192	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp)
56		nfv 1911	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(𝐽 ↾_t 𝐵) ∈ Comp
57		nfcv 2902	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐽
58		nfcv 2902	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 ↾_t
59	57, 58, 46	nfov 7460	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵)
60	59	nfel1 2919	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp
61	47	oveq2d 7446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑡 → (𝐽 ↾_t 𝐵) = (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵))
62	61	eleq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝐽 ↾_t 𝐵) ∈ Comp ↔ (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp))
63	56, 60, 62	cbvralw 3303	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp ↔ ∀𝑡 ∈ 𝐴 (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp)
64	55, 63	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∀𝑡 ∈ 𝐴 (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp)
65		ssun2 4188	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
66		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
67	65, 66	sstrid 4006	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → {𝑧} ⊆ 𝐴)
68	49	snss 4789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
69	67, 68	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → 𝑧 ∈ 𝐴)
70	54, 64, 69	rspcdva 3622	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp)
71		cmptop 23418	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top)
72		restrcl 23180	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top → (𝐽 ∈ V ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ V))
73	72	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top → ⦋𝑧 / 𝑥⦌𝐵 ∈ V)
74	70, 71, 73	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ⦋𝑧 / 𝑥⦌𝐵 ∈ V)
75	52, 74	eqeltrid 2842	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ V)
76		unexg 7761	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ V) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ V)
77	44, 75, 76	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ V)
78	39, 77	eqeltrid 2842	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V)
79		resttop 23183	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
80	38, 78, 79	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
81		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
82	81	restin 23189	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽 ↾_t (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽)))
83	38, 78, 82	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝐽 ↾_t (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽)))
84	83	unieqd 4924	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ∪ (𝐽 ↾_t (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽)))
85		inss2 4245	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
86		fiuncmp.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
87	85, 86	sseqtrri 4032	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ 𝑋
88	86	restuni 23185	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ 𝑋) → (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ∪ (𝐽 ↾_t (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽)))
89	38, 87, 88	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ∪ (𝐽 ↾_t (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽)))
90	84, 89	eqtr4d 2777	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽))
91	52	uneq2i 4174	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
92	39, 91	eqtri 2762	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
93	92	ineq1i 4223	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽)
94		indir 4291	. . . . . . . . . . . 12 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
95	93, 94	eqtri 2762	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
96	90, 95	eqtrdi 2790	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
97		inss1 4244	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ 𝑦 𝐵
98		ssun1 4187	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
99	98, 39	sseqtrri 4032	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
100	97, 99	sstri 4004	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
101	100	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
102		restabs 23188	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∧ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)))
103	38, 101, 78, 102	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)))
104	81	restin 23189	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) = (𝐽 ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)))
105	38, 44, 104	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) = (𝐽 ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)))
106	103, 105	eqtr4d 2777	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵))
107	106, 40	eqeltrd 2838	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)) ∈ Comp)
108		inss1 4244	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ⦋𝑧 / 𝑥⦌𝐵
109		ssun2 4188	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
110	109, 39	sseqtrri 4032	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
111	52, 110	eqsstrri 4030	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑧 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
112	108, 111	sstri 4004	. . . . . . . . . . . . . 14 ⊢ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
113	112	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
114		restabs 23188	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∧ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
115	38, 113, 78, 114	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
116	81	restin 23189	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ V) → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) = (𝐽 ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
117	38, 74, 116	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) = (𝐽 ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
118	115, 117	eqtr4d 2777	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)) = (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵))
119	118, 70	eqeltrd 2838	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)) ∈ Comp)
120		eqid 2734	. . . . . . . . . . 11 ⊢ ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
121	120	uncmp 23426	. . . . . . . . . 10 ⊢ ((((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top ∧ ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))) ∧ (((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)) ∈ Comp ∧ ((𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)
122	80, 96, 107, 119, 121	syl22anc 839	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)
123	122	exp32 420	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
124	123	a2d 29	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
125	37, 124	syl5 34	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → ((𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
126	125	a2i 14	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp)))
127	126	a1i 11	. . . 4 ⊢ (𝑦 ∈ Fin → (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))))
128	10, 16, 22, 28, 33, 127	findcard2 9202	. . 3 ⊢ (𝐴 ∈ Fin → ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)))
129	2, 128	mpcom 38	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp))
130	1, 129	mpi 20	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⦋csb 3907 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 {csn 4630 ∪ cuni 4911 ∪ ciun 4995 (class class class)co 7430 Fincfn 8983 ↾_t crest 17466 Topctop 22914 Compccmp 23409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-1o 8504 df-en 8984 df-dom 8985 df-fin 8987 df-fi 9448 df-rest 17468 df-topgen 17489 df-top 22915 df-topon 22932 df-bases 22968 df-cmp 23410

This theorem is referenced by: xkococnlem 23682

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