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Theorem fiuncmp 22907

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 4004	. 2 ⊢ 𝐴 ⊆ 𝐴
2		simp2 1137	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → 𝐴 ∈ Fin)
3		sseq1 4007	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		iuneq1 5013	. . . . . . . . 9 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
5		0iun 5066	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
6	4, 5	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∅)
7	6	oveq2d 7424	. . . . . . 7 ⊢ (𝑡 = ∅ → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∅))
8	7	eleq1d 2818	. . . . . 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 4007	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12		iuneq1 5013	. . . . . . . 8 ⊢ (𝑡 = 𝑦 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
13	12	oveq2d 7424	. . . . . . 7 ⊢ (𝑡 = 𝑦 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵))
14	13	eleq1d 2818	. . . . . 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 4007	. . . . . 6 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (𝑡 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
18		iuneq1 5013	. . . . . . . 8 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
19	18	oveq2d 7424	. . . . . . 7 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵))
20	19	eleq1d 2818	. . . . . 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 4007	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
24		iuneq1 5013	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
25	24	oveq2d 7424	. . . . . . 7 ⊢ (𝑡 = 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵))
26	25	eleq1d 2818	. . . . . 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 22672	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) = {∅})
30		0cmp 22897	. . . . . . 7 ⊢ {∅} ∈ Comp
31	29, 30	eqeltrdi 2841	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) ∈ Comp)
32	31	3ad2ant1 1133	. . . . 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 4172	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		id 22	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
36	34, 35	sstrid 3993	. . . . . . . 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 1191	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → 𝐽 ∈ Top)
39		iunxun 5097	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
40		simprr 771	. . . . . . . . . . . . . 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 22898	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top)
42		restrcl 22660	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top → (𝐽 ∈ V ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V))
43	42	simprd 496	. . . . . . . . . . . . . 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 2903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝐵
46		nfcsb1v 3918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑡 / 𝑥⦌𝐵
47		csbeq1a 3907	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → 𝐵 = ⦋𝑡 / 𝑥⦌𝐵)
48	45, 46, 47	cbviun 5039	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵
49		vex 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
50		csbeq1 3896	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
51	49, 50	iunxsn 5094	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
52	48, 51	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
53	50	oveq2d 7424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑧 → (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) = (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵))
54	53	eleq1d 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ((𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp ↔ (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp))
55		simpl3 1193	. . . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(𝐽 ↾_t 𝐵) ∈ Comp
57		nfcv 2903	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐽
58		nfcv 2903	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 ↾_t
59	57, 58, 46	nfov 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵)
60	59	nfel1 2919	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥(𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp
61	47	oveq2d 7424	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑡 → (𝐽 ↾_t 𝐵) = (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵))
62	61	eleq1d 2818	. . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . 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 4173	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
66		simprl 769	. . . . . . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . . . 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 22898	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top)
72		restrcl 22660	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top → (𝐽 ∈ V ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ V))
73	72	simprd 496	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 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 7735	. . . . . . . . . . . . 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 2837	. . . . . . . . . . 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 22663	. . . . . . . . . . 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 2732	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
82	81	restin 22669	. . . . . . . . . . . . . 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 4922	. . . . . . . . . . . 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 4229	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
86		fiuncmp.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
87	85, 86	sseqtrri 4019	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ 𝑋
88	86	restuni 22665	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 4160	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
92	39, 91	eqtri 2760	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
93	92	ineq1i 4208	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽)
94		indir 4275	. . . . . . . . . . . 12 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
95	93, 94	eqtri 2760	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
96	90, 95	eqtrdi 2788	. . . . . . . . . 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 4228	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ 𝑦 𝐵
98		ssun1 4172	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
99	98, 39	sseqtrri 4019	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
100	97, 99	sstri 3991	. . . . . . . . . . . . . 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 22668	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 22669	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 2833	. . . . . . . . . 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 4228	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ⦋𝑧 / 𝑥⦌𝐵
109		ssun2 4173	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
110	109, 39	sseqtrri 4019	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
111	52, 110	eqsstrri 4017	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑧 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
112	108, 111	sstri 3991	. . . . . . . . . . . . . 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 22668	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 22669	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 2833	. . . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
121	120	uncmp 22906	. . . . . . . . . 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 837	. . . . . . . . 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 421	. . . . . . . 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 9163	. . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⦋csb 3893 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 (class class class)co 7408 Fincfn 8938 ↾_t crest 17365 Topctop 22394 Compccmp 22889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-fin 8942 df-fi 9405 df-rest 17367 df-topgen 17388 df-top 22395 df-topon 22412 df-bases 22448 df-cmp 22890

This theorem is referenced by: xkococnlem 23162

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