Theorem alexsubALT 23995

Description: The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
alexsubALT.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
alexsubALT	⊢ (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))

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

Proof of Theorem alexsubALT

Dummy variables 𝑎 𝑏 𝑓 𝑡 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alexsubALT.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	alexsubALTlem1 23991	. 2 ⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
3	1	alexsubALTlem4 23994	. . . . 5 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) → ∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
4		velpw 4559	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 ⊆ 𝐽)
5		eleq2 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = ∪ 𝑐 → (𝑡 ∈ 𝑋 ↔ 𝑡 ∈ ∪ 𝑐))
6	5	3ad2ant3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ 𝑋 ↔ 𝑡 ∈ ∪ 𝑐))
7		eluni 4866	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ ∪ 𝑐 ↔ ∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ 𝑐))
8		ssel 3927	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑐 ⊆ 𝐽 → (𝑤 ∈ 𝑐 → 𝑤 ∈ 𝐽))
9		eleq2 2825	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤 ∈ 𝐽 ↔ 𝑤 ∈ (topGen‘(fi‘𝑥))))
10		tg2 22909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑤 ∈ (topGen‘(fi‘𝑥)) ∧ 𝑡 ∈ 𝑤) → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤))
11	10	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ (topGen‘(fi‘𝑥)) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤)))
12	9, 11	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (𝑤 ∈ 𝐽 → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤))))
13	8, 12	sylan9r 508	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → (𝑤 ∈ 𝑐 → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤))))
14	13	3impia 1117	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑤 ∈ 𝑐) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤)))
15		sseq2 3960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 = 𝑤 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑤))
16	15	rspcev 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑤 ∈ 𝑐 ∧ 𝑦 ⊆ 𝑤) → ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)
17	16	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ 𝑐 → (𝑦 ⊆ 𝑤 → ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧))
18	17	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑤 ∈ 𝑐) → (𝑦 ⊆ 𝑤 → ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧))
19	18	anim2d 612	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑤 ∈ 𝑐) → ((𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤) → (𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
20	19	reximdv 3151	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑤 ∈ 𝑐) → (∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑤) → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
21	14, 20	syld 47	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑤 ∈ 𝑐) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
22	21	3expia 1121	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → (𝑤 ∈ 𝑐 → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧))))
23	22	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → (𝑡 ∈ 𝑤 → (𝑤 ∈ 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧))))
24	23	impd 410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → ((𝑡 ∈ 𝑤 ∧ 𝑤 ∈ 𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
25	24	exlimdv 1934	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → (∃𝑤(𝑡 ∈ 𝑤 ∧ 𝑤 ∈ 𝑐) → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
26	7, 25	biimtrid 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽) → (𝑡 ∈ ∪ 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
27	26	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ ∪ 𝑐 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
28	6, 27	sylbid 240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ 𝑋 → ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
29		ssel 3927	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ⊆ 𝑧 → (𝑡 ∈ 𝑦 → 𝑡 ∈ 𝑧))
30		elunii 4868	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ 𝑧 ∧ 𝑧 ∈ 𝑐) → 𝑡 ∈ ∪ 𝑐)
31	30	expcom 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ 𝑐 → (𝑡 ∈ 𝑧 → 𝑡 ∈ ∪ 𝑐))
32	6	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ ∪ 𝑐 → 𝑡 ∈ 𝑋))
33	31, 32	sylan9r 508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑧 ∈ 𝑐) → (𝑡 ∈ 𝑧 → 𝑡 ∈ 𝑋))
34	29, 33	syl9r 78	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑧 ∈ 𝑐) → (𝑦 ⊆ 𝑧 → (𝑡 ∈ 𝑦 → 𝑡 ∈ 𝑋)))
35	34	rexlimdva 3137	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧 → (𝑡 ∈ 𝑦 → 𝑡 ∈ 𝑋)))
36	35	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ 𝑦 → (∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧 → 𝑡 ∈ 𝑋)))
37	36	impd 410	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ((𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧) → 𝑡 ∈ 𝑋))
38	37	rexlimdvw 3142	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧) → 𝑡 ∈ 𝑋))
39	28, 38	impbid 212	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ 𝑋 ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧)))
40		elunirab 4878	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ↔ ∃𝑦 ∈ (fi‘𝑥)(𝑡 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧))
41	39, 40	bitr4di 289	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑡 ∈ 𝑋 ↔ 𝑡 ∈ ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧}))
42	41	eqrdv 2734	. . . . . . . . . . . . . 14 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → 𝑋 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧})
43		ssrab2 4032	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ⊆ (fi‘𝑥)
44		fvex 6847	. . . . . . . . . . . . . . . . 17 ⊢ (fi‘𝑥) ∈ V
45	44	elpw2 5279	. . . . . . . . . . . . . . . 16 ⊢ ({𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∈ 𝒫 (fi‘𝑥) ↔ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ⊆ (fi‘𝑥))
46	43, 45	mpbir 231	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∈ 𝒫 (fi‘𝑥)
47		unieq 4874	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∪ 𝑎 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧})
48	47	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑋 = ∪ 𝑎 ↔ 𝑋 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧}))
49		pweq 4568	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → 𝒫 𝑎 = 𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧})
50	49	ineq1d 4171	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝒫 𝑎 ∩ Fin) = (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin))
51	50	rexeqdv 3297	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏 ↔ ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏))
52	48, 51	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ((𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ (𝑋 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏)))
53	52	rspcv 3572	. . . . . . . . . . . . . . 15 ⊢ ({𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∈ 𝒫 (fi‘𝑥) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝑋 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏)))
54	46, 53	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝑋 = ∪ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏))
55	42, 54	syl5com 31	. . . . . . . . . . . . 13 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → ∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏))
56		elfpw 9254	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin) ↔ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∧ 𝑏 ∈ Fin))
57		ssel 3927	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑡 ∈ 𝑏 → 𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧}))
58		sseq1 3959	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑡 → (𝑦 ⊆ 𝑧 ↔ 𝑡 ⊆ 𝑧))
59	58	rexbidv 3160	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑡 → (∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧))
60	59	elrab 3646	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ↔ (𝑡 ∈ (fi‘𝑥) ∧ ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧))
61	60	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧)
62	57, 61	syl6 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑡 ∈ 𝑏 → ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧))
63	62	ralrimiv 3127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∀𝑡 ∈ 𝑏 ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧)
64		sseq2 3960	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝑓‘𝑡) → (𝑡 ⊆ 𝑧 ↔ 𝑡 ⊆ (𝑓‘𝑡)))
65	64	ac6sfi 9184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ Fin ∧ ∀𝑡 ∈ 𝑏 ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧) → ∃𝑓(𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)))
66	65	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ Fin → (∀𝑡 ∈ 𝑏 ∃𝑧 ∈ 𝑐 𝑡 ⊆ 𝑧 → ∃𝑓(𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡))))
67	63, 66	syl5 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ Fin → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑓(𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡))))
68	67	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → ∃𝑓(𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡))))
69		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑓:𝑏⟶𝑐)
70		frn 6669	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓:𝑏⟶𝑐 → ran 𝑓 ⊆ 𝑐)
71	69, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ran 𝑓 ⊆ 𝑐)
72		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑏 ∈ Fin)
73		ffn 6662	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓:𝑏⟶𝑐 → 𝑓 Fn 𝑏)
74		dffn4 6752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 Fn 𝑏 ↔ 𝑓:𝑏–onto→ran 𝑓)
75	73, 74	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓:𝑏⟶𝑐 → 𝑓:𝑏–onto→ran 𝑓)
76	75	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) → 𝑓:𝑏–onto→ran 𝑓)
77	76	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑓:𝑏–onto→ran 𝑓)
78		fodomfi 9212	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 ∈ Fin ∧ 𝑓:𝑏–onto→ran 𝑓) → ran 𝑓 ≼ 𝑏)
79	72, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ran 𝑓 ≼ 𝑏)
80		domfi 9113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 ∈ Fin ∧ ran 𝑓 ≼ 𝑏) → ran 𝑓 ∈ Fin)
81	72, 79, 80	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ran 𝑓 ∈ Fin)
82	71, 81	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → (ran 𝑓 ⊆ 𝑐 ∧ ran 𝑓 ∈ Fin))
83		elin 3917	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ↔ (ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin))
84		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑐 ∈ V
85	84	elpw2 5279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ran 𝑓 ∈ 𝒫 𝑐 ↔ ran 𝑓 ⊆ 𝑐)
86	85	anbi1i 624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ran 𝑓 ∈ 𝒫 𝑐 ∧ ran 𝑓 ∈ Fin) ↔ (ran 𝑓 ⊆ 𝑐 ∧ ran 𝑓 ∈ Fin))
87	83, 86	bitr2i 276	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ran 𝑓 ⊆ 𝑐 ∧ ran 𝑓 ∈ Fin) ↔ ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
88	82, 87	sylib 218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin))
89		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑋 = ∪ 𝑏)
90		uniiun 5014	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ∪ 𝑏 = ∪ 𝑡 ∈ 𝑏 𝑡
91		simprlr 779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡))
92		ss2iun 4965	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡) → ∪ 𝑡 ∈ 𝑏 𝑡 ⊆ ∪ 𝑡 ∈ 𝑏 (𝑓‘𝑡))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∪ 𝑡 ∈ 𝑏 𝑡 ⊆ ∪ 𝑡 ∈ 𝑏 (𝑓‘𝑡))
94	90, 93	eqsstrid 3972	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∪ 𝑏 ⊆ ∪ 𝑡 ∈ 𝑏 (𝑓‘𝑡))
95		fniunfv 7193	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 Fn 𝑏 → ∪ 𝑡 ∈ 𝑏 (𝑓‘𝑡) = ∪ ran 𝑓)
96	69, 73, 95	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∪ 𝑡 ∈ 𝑏 (𝑓‘𝑡) = ∪ ran 𝑓)
97	94, 96	sseqtrd 3970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∪ 𝑏 ⊆ ∪ ran 𝑓)
98	89, 97	eqsstrd 3968	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑋 ⊆ ∪ ran 𝑓)
99		simpll2 1214	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑐 ⊆ 𝐽)
100	71, 99	sstrd 3944	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ran 𝑓 ⊆ 𝐽)
101		uniss 4871	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ran 𝑓 ⊆ 𝐽 → ∪ ran 𝑓 ⊆ ∪ 𝐽)
102	101, 1	sseqtrrdi 3975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ran 𝑓 ⊆ 𝐽 → ∪ ran 𝑓 ⊆ 𝑋)
103	100, 102	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∪ ran 𝑓 ⊆ 𝑋)
104	98, 103	eqssd 3951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → 𝑋 = ∪ ran 𝑓)
105		unieq 4874	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 = ran 𝑓 → ∪ 𝑑 = ∪ ran 𝑓)
106	105	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 = ran 𝑓 → (𝑋 = ∪ 𝑑 ↔ 𝑋 = ∪ ran 𝑓))
107	106	rspcev 3576	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ran 𝑓 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ ran 𝑓) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
108	88, 104, 107	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) ∧ ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) ∧ 𝑋 = ∪ 𝑏)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
109	108	exp32 420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) → ((𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
110	109	exlimdv 1934	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) → (∃𝑓(𝑓:𝑏⟶𝑐 ∧ ∀𝑡 ∈ 𝑏 𝑡 ⊆ (𝑓‘𝑡)) → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
111	68, 110	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) ∧ 𝑏 ∈ Fin) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
112	111	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑏 ∈ Fin → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
113	112	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} → (𝑏 ∈ Fin → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
114	113	impd 410	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ((𝑏 ⊆ {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∧ 𝑏 ∈ Fin) → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
115	56, 114	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin) → (𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
116	115	rexlimdv 3135	. . . . . . . . . . . . 13 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (∃𝑏 ∈ (𝒫 {𝑦 ∈ (fi‘𝑥) ∣ ∃𝑧 ∈ 𝑐 𝑦 ⊆ 𝑧} ∩ Fin)𝑋 = ∪ 𝑏 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
117	55, 116	syld 47	. . . . . . . . . . . 12 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
118	117	3exp 1119	. . . . . . . . . . 11 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
119	118	com34 91	. . . . . . . . . 10 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (𝑐 ⊆ 𝐽 → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
120	119	com23 86	. . . . . . . . 9 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
121	4, 120	syl7bi 255	. . . . . . . 8 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
122	121	ralrimdv 3134	. . . . . . 7 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
123		fibas 22921	. . . . . . . . 9 ⊢ (fi‘𝑥) ∈ TopBases
124		tgcl 22913	. . . . . . . . 9 ⊢ ((fi‘𝑥) ∈ TopBases → (topGen‘(fi‘𝑥)) ∈ Top)
125	123, 124	ax-mp 5	. . . . . . . 8 ⊢ (topGen‘(fi‘𝑥)) ∈ Top
126		eleq1 2824	. . . . . . . 8 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (𝐽 ∈ Top ↔ (topGen‘(fi‘𝑥)) ∈ Top))
127	125, 126	mpbiri 258	. . . . . . 7 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → 𝐽 ∈ Top)
128	122, 127	jctild 525	. . . . . 6 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
129	1	iscmp 23332	. . . . . 6 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
130	128, 129	imbitrrdi 252	. . . . 5 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) → 𝐽 ∈ Comp))
131	3, 130	syld 47	. . . 4 ⊢ (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) → 𝐽 ∈ Comp))
132	131	imp 406	. . 3 ⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp)
133	132	exlimiv 1931	. 2 ⊢ (∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp)
134	2, 133	impbii 209	1 ⊢ (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863  ∪ ciun 4946   class class class wbr 5098  ran crn 5625   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492   ≼ cdom 8881  Fincfn 8883  ficfi 9313  topGenctg 17357  Topctop 22837  TopBasesctb 22889  Compccmp 23330

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-ac2 10373

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-rpss 7668  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9314  df-card 9851  df-ac 10026  df-topgen 17363  df-top 22838  df-bases 22890  df-cmp 23331

This theorem is referenced by: (None)