Theorem cmpsub 21203

Description: Two equivalent ways of describing a compact subset of a topological space. Inspired by Sue E. Goodman's Beginning Topology. (Contributed by Jeff Hankins, 22-Jun-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Hypothesis

Ref	Expression
cmpsub.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cmpsub	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))

Distinct variable groups:   𝑐,𝑑,𝐽   𝑆,𝑐,𝑑   𝑋,𝑐,𝑑

Proof of Theorem cmpsub

Dummy variables 𝑥 𝑦 𝑓 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
2	1	iscmp 21191	. . 3 ⊢ ((𝐽 ↾t 𝑆) ∈ Comp ↔ ((𝐽 ↾t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
3		id 22	. . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ⊆ 𝑋)
4		cmpsub.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
5	4	topopn 20711	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
6		ssexg 4804	. . . . . 6 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
7	3, 5, 6	syl2anr 495	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
8		resttop 20964	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
9	7, 8	syldan 487	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
10		ibar 525	. . . . 5 ⊢ ((𝐽 ↾t 𝑆) ∈ Top → (∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) ↔ ((𝐽 ↾t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))))
11	10	bicomd 213	. . . 4 ⊢ ((𝐽 ↾t 𝑆) ∈ Top → (((𝐽 ↾t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
12	9, 11	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((𝐽 ↾t 𝑆) ∈ Top ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
13	2, 12	syl5bb 272	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
14		vex 3203	. . . . . . . . . . 11 ⊢ 𝑡 ∈ V
15		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝑦 ∩ 𝑆) ↔ 𝑡 = (𝑦 ∩ 𝑆)))
16	15	rexbidv 3052	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆) ↔ ∃𝑦 ∈ 𝑐 𝑡 = (𝑦 ∩ 𝑆)))
17	14, 16	elab 3350	. . . . . . . . . 10 ⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ ∃𝑦 ∈ 𝑐 𝑡 = (𝑦 ∩ 𝑆))
18		selpw 4165	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 ⊆ 𝐽)
19		ssel2 3598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ⊆ 𝐽 ∧ 𝑦 ∈ 𝑐) → 𝑦 ∈ 𝐽)
20		ineq1 3807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑦 → (𝑑 ∩ 𝑆) = (𝑦 ∩ 𝑆))
21	20	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑦 → (𝑡 = (𝑑 ∩ 𝑆) ↔ 𝑡 = (𝑦 ∩ 𝑆)))
22	21	rspcev 3309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑡 = (𝑦 ∩ 𝑆)) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆))
23	22	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐽 → (𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆)))
24	19, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ⊆ 𝐽 ∧ 𝑦 ∈ 𝑐) → (𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆)))
25	24	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝑐 ⊆ 𝐽 → (𝑦 ∈ 𝑐 → (𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆))))
26	18, 25	sylbi 207	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝒫 𝐽 → (𝑦 ∈ 𝑐 → (𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆))))
27	26	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑦 ∈ 𝑐 → (𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆))))
28	27	rexlimdv 3030	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∃𝑦 ∈ 𝑐 𝑡 = (𝑦 ∩ 𝑆) → ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆)))
29		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → 𝐽 ∈ Top)
30	4	sseq2i 3630	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ ∪ 𝐽)
31		uniexg 6955	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
32		ssexg 4804	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ∈ V) → 𝑆 ∈ V)
33	31, 32	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ∪ 𝐽 ∧ 𝐽 ∈ Top) → 𝑆 ∈ V)
34	33	ancoms 469	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ∈ V)
35	30, 34	sylan2b 492	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
36	35	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → 𝑆 ∈ V)
37		elrest 16088	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝑡 ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆)))
38	29, 36, 37	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑡 ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑑 ∈ 𝐽 𝑡 = (𝑑 ∩ 𝑆)))
39	28, 38	sylibrd 249	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∃𝑦 ∈ 𝑐 𝑡 = (𝑦 ∩ 𝑆) → 𝑡 ∈ (𝐽 ↾t 𝑆)))
40	17, 39	syl5bi 232	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑡 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → 𝑡 ∈ (𝐽 ↾t 𝑆)))
41	40	ssrdv 3609	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ⊆ (𝐽 ↾t 𝑆))
42		vex 3203	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
43	42	abrexex 7141	. . . . . . . . 9 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∈ V
44	43	elpw 4164	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∈ 𝒫 (𝐽 ↾t 𝑆) ↔ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ⊆ (𝐽 ↾t 𝑆))
45	41, 44	sylibr 224	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∈ 𝒫 (𝐽 ↾t 𝑆))
46		unieq 4444	. . . . . . . . . 10 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∪ 𝑠 = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)})
47	46	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → (∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 ↔ ∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)}))
48		pweq 4161	. . . . . . . . . . 11 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → 𝒫 𝑠 = 𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)})
49	48	ineq1d 3813	. . . . . . . . . 10 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → (𝒫 𝑠 ∩ Fin) = (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin))
50	49	rexeqdv 3145	. . . . . . . . 9 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → (∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡 ↔ ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))
51	47, 50	imbi12d 334	. . . . . . . 8 ⊢ (𝑠 = {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ((∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) ↔ (∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
52	51	rspcva 3307	. . . . . . 7 ⊢ (({𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∈ 𝒫 (𝐽 ↾t 𝑆) ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)) → (∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))
53	45, 52	sylan 488	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)) → (∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))
54	53	ex 450	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → (∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
55	4	restuni 20966	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
56	55	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
57		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
58	57	inex1 4799	. . . . . . . . . . . . 13 ⊢ (𝑦 ∩ 𝑆) ∈ V
59	58	dfiun2 4554	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ 𝑐 (𝑦 ∩ 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)}
60		incom 3805	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∩ 𝑆) = (𝑆 ∩ 𝑦)
61	60	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ 𝑦 ∈ 𝑐) → (𝑦 ∩ 𝑆) = (𝑆 ∩ 𝑦))
62	61	iuneq2dv 4542	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ∪ 𝑦 ∈ 𝑐 (𝑦 ∩ 𝑆) = ∪ 𝑦 ∈ 𝑐 (𝑆 ∩ 𝑦))
63	59, 62	syl5eqr 2670	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} = ∪ 𝑦 ∈ 𝑐 (𝑆 ∩ 𝑦))
64		iunin2 4584	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ 𝑐 (𝑆 ∩ 𝑦) = (𝑆 ∩ ∪ 𝑦 ∈ 𝑐 𝑦)
65		uniiun 4573	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑐 = ∪ 𝑦 ∈ 𝑐 𝑦
66	65	eqcomi 2631	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ 𝑐 𝑦 = ∪ 𝑐
67	66	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ∪ 𝑦 ∈ 𝑐 𝑦 = ∪ 𝑐)
68	67	ineq2d 3814	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (𝑆 ∩ ∪ 𝑦 ∈ 𝑐 𝑦) = (𝑆 ∩ ∪ 𝑐))
69		incom 3805	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∩ ∪ 𝑐) = (∪ 𝑐 ∩ 𝑆)
70		sseqin2 3817	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ ∪ 𝑐 ↔ (∪ 𝑐 ∩ 𝑆) = 𝑆)
71	70	biimpi 206	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ ∪ 𝑐 → (∪ 𝑐 ∩ 𝑆) = 𝑆)
72	69, 71	syl5eq 2668	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ∪ 𝑐 → (𝑆 ∩ ∪ 𝑐) = 𝑆)
73	72	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (𝑆 ∩ ∪ 𝑐) = 𝑆)
74	68, 73	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (𝑆 ∩ ∪ 𝑦 ∈ 𝑐 𝑦) = 𝑆)
75	64, 74	syl5eq 2668	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ∪ 𝑦 ∈ 𝑐 (𝑆 ∩ 𝑦) = 𝑆)
76	63, 75	eqtr2d 2657	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → 𝑆 = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)})
77	56, 76	eqeq12d 2637	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (𝑆 = 𝑆 ↔ ∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)}))
78	56	eqeq1d 2624	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (𝑆 = ∪ 𝑡 ↔ ∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))
79	78	rexbidv 3052	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡 ↔ ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))
80	77, 79	imbi12d 334	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ((𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡) ↔ (∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
81		eqid 2622	. . . . . . . . . 10 ⊢ 𝑆 = 𝑆
82	81	a1bi 352	. . . . . . . . 9 ⊢ (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡 ↔ (𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡))
83		elin 3796	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin) ↔ (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∧ 𝑡 ∈ Fin))
84		selpw 4165	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ 𝑡 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)})
85		dfss3 3592	. . . . . . . . . . . . . 14 ⊢ (𝑡 ⊆ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ ∀𝑠 ∈ 𝑡 𝑠 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)})
86		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑠 ∈ V
87		eqeq1 2626	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑠 → (𝑥 = (𝑦 ∩ 𝑆) ↔ 𝑠 = (𝑦 ∩ 𝑆)))
88	87	rexbidv 3052	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆) ↔ ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆)))
89	86, 88	elab 3350	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆))
90	89	ralbii 2980	. . . . . . . . . . . . . 14 ⊢ (∀𝑠 ∈ 𝑡 𝑠 ∈ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ ∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆))
91	84, 85, 90	3bitri 286	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ↔ ∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆))
92	91	anbi1i 731	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∧ 𝑡 ∈ Fin) ↔ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin))
93	83, 92	bitri 264	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin) ↔ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin))
94		ineq1 3807	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑓‘𝑠) → (𝑦 ∩ 𝑆) = ((𝑓‘𝑠) ∩ 𝑆))
95	94	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑓‘𝑠) → (𝑠 = (𝑦 ∩ 𝑆) ↔ 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)))
96	95	ac6sfi 8204	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Fin ∧ ∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆)) → ∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)))
97	96	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin) → ∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)))
98	97	adantl 482	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → ∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)))
99		frn 6053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑡⟶𝑐 → ran 𝑓 ⊆ 𝑐)
100	99	ad2antrl 764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ran 𝑓 ⊆ 𝑐)
101		vex 3203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑓 ∈ V
102	101	rnex 7100	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran 𝑓 ∈ V
103	102	elpw 4164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ran 𝑓 ∈ 𝒫 𝑐 ↔ ran 𝑓 ⊆ 𝑐)
104	100, 103	sylibr 224	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ran 𝑓 ∈ 𝒫 𝑐)
105		simprr 796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → 𝑡 ∈ Fin)
106	105	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → 𝑡 ∈ Fin)
107		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓:𝑡⟶𝑐 → 𝑓 Fn 𝑡)
108		dffn4 6121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 Fn 𝑡 ↔ 𝑓:𝑡–onto→ran 𝑓)
109	107, 108	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:𝑡⟶𝑐 → 𝑓:𝑡–onto→ran 𝑓)
110		fodomfi 8239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡–onto→ran 𝑓) → ran 𝑓 ≼ 𝑡)
111	109, 110	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ Fin ∧ 𝑓:𝑡⟶𝑐) → ran 𝑓 ≼ 𝑡)
112	111	adantll 750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin) ∧ 𝑓:𝑡⟶𝑐) → ran 𝑓 ≼ 𝑡)
113	112	adantll 750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑓:𝑡⟶𝑐) → ran 𝑓 ≼ 𝑡)
114	113	ad2ant2r 783	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ran 𝑓 ≼ 𝑡)
115		domfi 8181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ Fin ∧ ran 𝑓 ≼ 𝑡) → ran 𝑓 ∈ Fin)
116	106, 114, 115	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ran 𝑓 ∈ Fin)
117	104, 116	elind 3798	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
118		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑢 → 𝑠 = 𝑢)
119		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑠 = 𝑢 → (𝑓‘𝑠) = (𝑓‘𝑢))
120	119	ineq1d 3813	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑢 → ((𝑓‘𝑠) ∩ 𝑆) = ((𝑓‘𝑢) ∩ 𝑆))
121	118, 120	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑢 → (𝑠 = ((𝑓‘𝑠) ∩ 𝑆) ↔ 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)))
122	121	rspccv 3306	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆) → (𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)))
123		pm2.27 42	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑢 ∈ 𝑡 → ((𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)) → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)))
124		inss1 3833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓‘𝑢) ∩ 𝑆) ⊆ (𝑓‘𝑢)
125		sseq1 3626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑢 = ((𝑓‘𝑢) ∩ 𝑆) → (𝑢 ⊆ (𝑓‘𝑢) ↔ ((𝑓‘𝑢) ∩ 𝑆) ⊆ (𝑓‘𝑢)))
126	124, 125	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑢 = ((𝑓‘𝑢) ∩ 𝑆) → 𝑢 ⊆ (𝑓‘𝑢))
127		ssel 3597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑢 ⊆ (𝑓‘𝑢) → (𝑤 ∈ 𝑢 → 𝑤 ∈ (𝑓‘𝑢)))
128	127	a1dd 50	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑢 ⊆ (𝑓‘𝑢) → (𝑤 ∈ 𝑢 → (𝑓:𝑡⟶𝑐 → 𝑤 ∈ (𝑓‘𝑢))))
129	126, 128	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑢 = ((𝑓‘𝑢) ∩ 𝑆) → (𝑤 ∈ 𝑢 → (𝑓:𝑡⟶𝑐 → 𝑤 ∈ (𝑓‘𝑢))))
130	129	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑢 ∈ 𝑡 → (𝑢 = ((𝑓‘𝑢) ∩ 𝑆) → (𝑤 ∈ 𝑢 → (𝑓:𝑡⟶𝑐 → 𝑤 ∈ (𝑓‘𝑢)))))
131	130	3imp 1256	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑢 ∈ 𝑡 ∧ 𝑢 = ((𝑓‘𝑢) ∩ 𝑆) ∧ 𝑤 ∈ 𝑢) → (𝑓:𝑡⟶𝑐 → 𝑤 ∈ (𝑓‘𝑢)))
132		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓 Fn 𝑡 ∧ 𝑢 ∈ 𝑡) → (𝑓‘𝑢) ∈ ran 𝑓)
133	132	expcom 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑢 ∈ 𝑡 → (𝑓 Fn 𝑡 → (𝑓‘𝑢) ∈ ran 𝑓))
134	133	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑢 ∈ 𝑡 ∧ 𝑢 = ((𝑓‘𝑢) ∩ 𝑆) ∧ 𝑤 ∈ 𝑢) → (𝑓 Fn 𝑡 → (𝑓‘𝑢) ∈ ran 𝑓))
135	107, 134	syl5 34	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑢 ∈ 𝑡 ∧ 𝑢 = ((𝑓‘𝑢) ∩ 𝑆) ∧ 𝑤 ∈ 𝑢) → (𝑓:𝑡⟶𝑐 → (𝑓‘𝑢) ∈ ran 𝑓))
136	131, 135	jcad 555	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑢 ∈ 𝑡 ∧ 𝑢 = ((𝑓‘𝑢) ∩ 𝑆) ∧ 𝑤 ∈ 𝑢) → (𝑓:𝑡⟶𝑐 → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))
137	136	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑢 ∈ 𝑡 → (𝑢 = ((𝑓‘𝑢) ∩ 𝑆) → (𝑤 ∈ 𝑢 → (𝑓:𝑡⟶𝑐 → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))))
138	123, 137	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 ∈ 𝑡 → ((𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)) → (𝑤 ∈ 𝑢 → (𝑓:𝑡⟶𝑐 → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))))
139	138	com3r 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ 𝑢 → (𝑢 ∈ 𝑡 → ((𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)) → (𝑓:𝑡⟶𝑐 → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))))
140	139	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → ((𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)) → (𝑓:𝑡⟶𝑐 → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓))))
141	140	com3l 89	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆)) → (𝑓:𝑡⟶𝑐 → ((𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓))))
142	141	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:𝑡⟶𝑐 ∧ (𝑢 ∈ 𝑡 → 𝑢 = ((𝑓‘𝑢) ∩ 𝑆))) → ((𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))
143	122, 142	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → ((𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))
144		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓‘𝑢) ∈ V
145		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ (𝑓‘𝑢)))
146		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = (𝑓‘𝑢) → (𝑣 ∈ ran 𝑓 ↔ (𝑓‘𝑢) ∈ ran 𝑓))
147	145, 146	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = (𝑓‘𝑢) → ((𝑤 ∈ 𝑣 ∧ 𝑣 ∈ ran 𝑓) ↔ (𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓)))
148	144, 147	spcev 3300	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 ∈ (𝑓‘𝑢) ∧ (𝑓‘𝑢) ∈ ran 𝑓) → ∃𝑣(𝑤 ∈ 𝑣 ∧ 𝑣 ∈ ran 𝑓))
149	143, 148	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → ((𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → ∃𝑣(𝑤 ∈ 𝑣 ∧ 𝑣 ∈ ran 𝑓)))
150	149	exlimdv 1861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → (∃𝑢(𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡) → ∃𝑣(𝑤 ∈ 𝑣 ∧ 𝑣 ∈ ran 𝑓)))
151		eluni 4439	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ∪ 𝑡 ↔ ∃𝑢(𝑤 ∈ 𝑢 ∧ 𝑢 ∈ 𝑡))
152		eluni 4439	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ∪ ran 𝑓 ↔ ∃𝑣(𝑤 ∈ 𝑣 ∧ 𝑣 ∈ ran 𝑓))
153	150, 151, 152	3imtr4g 285	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → (𝑤 ∈ ∪ 𝑡 → 𝑤 ∈ ∪ ran 𝑓))
154	153	ssrdv 3609	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → ∪ 𝑡 ⊆ ∪ ran 𝑓)
155	154	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → ∪ 𝑡 ⊆ ∪ ran 𝑓)
156		sseq1 3626	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 = ∪ 𝑡 → (𝑆 ⊆ ∪ ran 𝑓 ↔ ∪ 𝑡 ⊆ ∪ ran 𝑓))
157	156	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → (𝑆 ⊆ ∪ ran 𝑓 ↔ ∪ 𝑡 ⊆ ∪ ran 𝑓))
158	155, 157	mpbird 247	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → 𝑆 ⊆ ∪ ran 𝑓)
159	117, 158	jca 554	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) ∧ (𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆))) → (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓))
160	159	ex 450	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) → ((𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓)))
161	160	eximdv 1846	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) ∧ 𝑆 = ∪ 𝑡) → (∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓)))
162	161	ex 450	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → (𝑆 = ∪ 𝑡 → (∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓))))
163	162	com23 86	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → (∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → (𝑆 = ∪ 𝑡 → ∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓))))
164		unieq 4444	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ran 𝑓 → ∪ 𝑑 = ∪ ran 𝑓)
165	164	sseq2d 3633	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ran 𝑓 → (𝑆 ⊆ ∪ 𝑑 ↔ 𝑆 ⊆ ∪ ran 𝑓))
166	165	rspcev 3309	. . . . . . . . . . . . . 14 ⊢ ((ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)
167	166	exlimiv 1858	. . . . . . . . . . . . 13 ⊢ (∃𝑓(ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑆 ⊆ ∪ ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)
168	163, 167	syl8 76	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → (∃𝑓(𝑓:𝑡⟶𝑐 ∧ ∀𝑠 ∈ 𝑡 𝑠 = ((𝑓‘𝑠) ∩ 𝑆)) → (𝑆 = ∪ 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
169	98, 168	mpd 15	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ (∀𝑠 ∈ 𝑡 ∃𝑦 ∈ 𝑐 𝑠 = (𝑦 ∩ 𝑆) ∧ 𝑡 ∈ Fin)) → (𝑆 = ∪ 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))
170	93, 169	sylan2b 492	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) ∧ 𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)) → (𝑆 = ∪ 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))
171	170	rexlimdva 3031	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → (∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))
172	82, 171	syl5bir 233	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ((𝑆 = 𝑆 → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)𝑆 = ∪ 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))
173	80, 172	sylbird 250	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) ∧ 𝑆 ⊆ ∪ 𝑐) → ((∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑))
174	173	ex 450	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (𝑆 ⊆ ∪ 𝑐 → ((∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
175	174	com23 86	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → ((∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} → ∃𝑡 ∈ (𝒫 {𝑥 ∣ ∃𝑦 ∈ 𝑐 𝑥 = (𝑦 ∩ 𝑆)} ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → (𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
176	54, 175	syld 47	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑐 ∈ 𝒫 𝐽) → (∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → (𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
177	176	ralrimdva 2969	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) → ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
178	4	cmpsublem 21202	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡)))
179	177, 178	impbid 202	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪ (𝐽 ↾t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡) ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
180	13, 179	bitrd 268	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   ≼ cdom 7953  Fincfn 7955   ↾_t crest 16081  Topctop 20698  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-rep 4771  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-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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190

This theorem is referenced by:  cmpcld  21205  uncmp  21206  hauscmplem  21209  1stckgenlem  21356  icccmp  22628  bndth  22757  ovolicc2  23290  stoweidlem50  40267  stoweidlem57  40274