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Theorem fiuncmp 23737

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 3935	. 2 ⊢ 𝐴 ⊆ 𝐴
2		simp2 1155	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → 𝐴 ∈ Fin)
3		sseq1 3938	. . . . . 6 ⊢ (𝑡 = ∅ → (𝑡 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		iuneq1 4971	. . . . . . . . 9 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
5		0iun 5024	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
6	4, 5	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑡 = ∅ → ∪ 𝑥 ∈ 𝑡 𝐵 = ∅)
7	6	oveq2d 7446	. . . . . . 7 ⊢ (𝑡 = ∅ → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) = (𝐽 ↾_t ∅))
8	7	eleq1d 2849	. . . . . 6 ⊢ (𝑡 = ∅ → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∅) ∈ Comp))
9	3, 8	imbi12d 347	. . . . 5 ⊢ (𝑡 = ∅ → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (∅ ⊆ 𝐴 → (𝐽 ↾_t ∅) ∈ Comp)))
10	9	imbi2d 343	. . . 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 3938	. . . . . 6 ⊢ (𝑡 = 𝑦 → (𝑡 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12		iuneq1 4971	. . . . . . . 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 2849	. . . . . 6 ⊢ (𝑡 = 𝑦 → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp))
15	11, 14	imbi12d 347	. . . . 5 ⊢ (𝑡 = 𝑦 → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)))
16	15	imbi2d 343	. . . 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 3938	. . . . . 6 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → (𝑡 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
18		iuneq1 4971	. . . . . . . 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 2849	. . . . . 6 ⊢ (𝑡 = (𝑦 ∪ {𝑧}) → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Comp))
21	17, 20	imbi12d 347	. . . . 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 343	. . . 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 3938	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
24		iuneq1 4971	. . . . . . . 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 2849	. . . . . 6 ⊢ (𝑡 = 𝐴 → ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp ↔ (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp))
27	23, 26	imbi12d 347	. . . . 5 ⊢ (𝑡 = 𝐴 → ((𝑡 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑡 𝐵) ∈ Comp) ↔ (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)))
28	27	imbi2d 343	. . . 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 23502	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) = {∅})
30		0cmp 23727	. . . . . . 7 ⊢ {∅} ∈ Comp
31	29, 30	eqeltrdi 2872	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝐽 ↾_t ∅) ∈ Comp)
32	31	3ad2ant1 1151	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐽 ↾_t ∅) ∈ Comp)
33	32	a1d 26	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (∅ ⊆ 𝐴 → (𝐽 ↾_t ∅) ∈ Comp))
34		ssun1 4179	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
35		id 23	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
36	34, 35	sstrid 3924	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑦 ⊆ 𝐴)
37	36	imim1i 64	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp))
38		simpl1 1210	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp)) → 𝐽 ∈ Top)
39		iunxun 5057	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
40		simprr 785	. . . . . . . . . . . . . 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 23728	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Comp → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top)
42		restrcl 23490	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top → (𝐽 ∈ V ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V))
43	42	simprd 501	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) ∈ Top → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V)
44	40, 41, 43	3syl 19	. . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡𝐵
46		nfcsb1v 3875	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑡 / 𝑥⦌𝐵
47		csbeq1a 3865	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → 𝐵 = ⦋𝑡 / 𝑥⦌𝐵)
48	45, 46, 47	cbviun 4996	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵
49		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
50		csbeq1 3854	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
51	49, 50	iunxsn 5054	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑡 ∈ {𝑧}⦋𝑡 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
52	48, 51	eqtri 2786	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → ((𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp ↔ (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp))
55		simpl3 1212	. . . . . . . . . . . . . . . . 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 1948	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(𝐽 ↾_t 𝐵) ∈ Comp
57		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐽
58		nfcv 2926	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥 ↾_t
59	57, 58, 46	nfov 7460	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵)
60	59	nfel1 2942	. . . . . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → ((𝐽 ↾_t 𝐵) ∈ Comp ↔ (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp))
63	56, 60, 62	cbvralw 3308	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp ↔ ∀𝑡 ∈ 𝐴 (𝐽 ↾_t ⦋𝑡 / 𝑥⦌𝐵) ∈ Comp)
64	55, 63	sylib 221	. . . . . . . . . . . . . . . 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 4180	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
66		simprl 783	. . . . . . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . . . 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 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
69	67, 68	sylibr 237	. . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . 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 23728	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Comp → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top)
72		restrcl 23490	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top → (𝐽 ∈ V ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ V))
73	72	simprd 501	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) ∈ Top → ⦋𝑧 / 𝑥⦌𝐵 ∈ V)
74	70, 71, 73	3syl 19	. . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . 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 7770	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ V ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ V) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ V)
77	44, 75, 76	syl2anc 596	. . . . . . . . . . . 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 2868	. . . . . . . . . . 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 23493	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ V) → (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ Top)
80	38, 78, 79	syl2anc 596	. . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝐽 = ∪ 𝐽
82	81	restin 23499	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 4883	. . . . . . . . . . . 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 4137	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
86		fiuncmp.1	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ∪ 𝐽
87	85, 86	sseqtrri 3962	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) ⊆ 𝑋
88	86	restuni 23495	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 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 2801	. . . . . . . . . . 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 4167	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
92	39, 91	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵)
93	92	ineq1i 4116	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽)
94		indir 4234	. . . . . . . . . . . 12 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∪ ⦋𝑧 / 𝑥⦌𝐵) ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
95	93, 94	eqtri 2786	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∩ ∪ 𝐽) = ((∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ∪ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽))
96	90, 95	eqtrdi 2814	. . . . . . . . . 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 4136	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽) ⊆ ∪ 𝑥 ∈ 𝑦 𝐵
98		ssun1 4179	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
99	98, 39	sseqtrri 3962	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ 𝑦 𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
100	97, 99	sstri 3922	. . . . . . . . . . . . . 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 23498	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 23499	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ V) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝑦 𝐵) = (𝐽 ↾_t (∪ 𝑥 ∈ 𝑦 𝐵 ∩ ∪ 𝐽)))
105	38, 44, 104	syl2anc 596	. . . . . . . . . . . 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 2801	. . . . . . . . . . 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 2864	. . . . . . . . . 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 4136	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽) ⊆ ⦋𝑧 / 𝑥⦌𝐵
109		ssun2 4180	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
110	109, 39	sseqtrri 3962	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
111	52, 110	eqsstrri 3960	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑧 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵
112	108, 111	sstri 3922	. . . . . . . . . . . . . 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 23498	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 23499	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ V) → (𝐽 ↾_t ⦋𝑧 / 𝑥⦌𝐵) = (𝐽 ↾_t (⦋𝑧 / 𝑥⦌𝐵 ∩ ∪ 𝐽)))
117	38, 74, 116	syl2anc 596	. . . . . . . . . . . 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 2801	. . . . . . . . . . 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 2864	. . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) = ∪ (𝐽 ↾_t ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
121	120	uncmp 23736	. . . . . . . . . 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 852	. . . . . . . . 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 426	. . . . . . . 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 30	. . . . . . 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 35	. . . . . 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 15	. . . . 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 9194	. . 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 39	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐴 ⊆ 𝐴 → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp))
130	1, 129	mpi 21	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 (𝐽 ↾_t 𝐵) ∈ Comp) → (𝐽 ↾_t ∪ 𝑥 ∈ 𝐴 𝐵) ∈ Comp)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∀wral 3080 Vcvv 3454 ⦋csb 3851 ⊆ wss 3900 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 ∪ cuni 4870 ∪ ciun 4954 (class class class)co 7430 Fincfn 8987 ↾_t crest 17606 Topctop 23226 Compccmp 23719

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-1o 8490 df-en 8988 df-dom 8989 df-fin 8991 df-fi 9416 df-rest 17608 df-topgen 17629 df-top 23227 df-topon 23244 df-bases 23279 df-cmp 23720

This theorem is used by: xkococnlem 23993

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