Theorem locfincmp 23652

Description: For a compact space, the locally finite covers are precisely the finite covers. Sadly, this property does not properly characterize all compact spaces. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
locfincmp.1	⊢ 𝑋 = ∪ 𝐽
locfincmp.2	⊢ 𝑌 = ∪ 𝐶

Assertion

Ref	Expression
locfincmp	⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Proof of Theorem locfincmp

Dummy variables 𝑜 𝑐 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		locfincmp.1	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
2	1	locfinnei 23649	. . . . . . . . 9 ⊢ ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
3	2	ralrimiva 3163	. . . . . . . 8 ⊢ (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
4	1	cmpcov2 23516	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
5	3, 4	sylan2 604	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
6		elfpw 9311	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin))
7		simplrr 789	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → 𝑐 ∈ Fin)
8		eldifsn 4756	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅))
9		ineq1 4172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (𝑠 ∩ 𝑜) = (𝑥 ∩ 𝑜))
10	9	neeq1d 3023	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → ((𝑠 ∩ 𝑜) ≠ ∅ ↔ (𝑥 ∩ 𝑜) ≠ ∅))
11		simplrl 788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ 𝐶)
12		simplrr 789	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑥)
13		simprr 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑜)
14		inelcm 4429	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ 𝑜) → (𝑥 ∩ 𝑜) ≠ ∅)
15	12, 13, 14	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → (𝑥 ∩ 𝑜) ≠ ∅)
16	10, 11, 15	elrabd 3659	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
17		elunii 4879	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
18		locfincmp.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑌 = ∪ 𝐶
19	17, 18	eleqtrrdi 2880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ 𝑌)
20	19	ancoms 463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
21	20	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑌)
22	1, 18	locfinbas 23648	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
23	22	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
24	23	ad3antrrr 742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = 𝑌)
25	21, 24	eleqtrrd 2872	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
26		simplr 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = ∪ 𝑐)
27	25, 26	eleqtrd 2871	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝑐)
28		eluni2 4878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ∪ 𝑐 ↔ ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
29	27, 28	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
30	16, 29	reximddv 3187	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
31	30	expr 461	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ 𝑥 ∈ 𝐶) → (𝑦 ∈ 𝑥 → ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
32	31	exlimdv 1960	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ 𝑥 ∈ 𝐶) → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
33		n0 4313	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
34		eliun 4962	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ↔ ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
35	32, 33, 34	3imtr4g 299	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ 𝑥 ∈ 𝐶) → (𝑥 ≠ ∅ → 𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
36	35	expimpd 458	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → ((𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅) → 𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
37	8, 36	biimtrid 245	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
38	37	ssrdv 3949	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
39		iunfi 9300	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ Fin ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin)
40	39	ex 417	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ Fin → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
41		ssfi 9157	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
42	41	expcom 418	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} → (∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
43	40, 42	sylan9 516	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
44	7, 38, 43	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
45	44	expimpd 458	. . . . . . . . 9 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
46	6, 45	sylan2b 605	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
47	46	rexlimdva 3172	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
48	5, 47	mpd 16	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∖ {∅}) ∈ Fin)
49		snfi 9040	. . . . . 6 ⊢ {∅} ∈ Fin
50		unfi 9155	. . . . . 6 ⊢ (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
51	48, 49, 50	sylancl 597	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52		ssun1 4137	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {∅})
53		undif1 4440	. . . . . 6 ⊢ ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
54	52, 53	sseqtrri 3992	. . . . 5 ⊢ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55		ssfi 9157	. . . . 5 ⊢ ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
56	51, 54, 55	sylancl 597	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
57	56, 23	jca 520	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
58	57	ex 417	. 2 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59		cmptop 23521	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
60	1, 18	finlocfin 23646	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61	60	3expib 1138	. . 3 ⊢ (𝐽 ∈ Top → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
62	59, 61	syl 18	. 2 ⊢ (𝐽 ∈ Comp → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
63	58, 62	impbid 215	1 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3422   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4565  {csn 4592  ∪ cuni 4874  ∪ ciun 4958  ‘cfv 6537  Fincfn 8943  Topctop 23019  Compccmp 23512  LocFinclocfin 23630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-en 8944  df-fin 8947  df-top 23020  df-cmp 23513  df-locfin 23633

This theorem is referenced by: (None)