Theorem locfincmp 23472

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 23469	. . . . . . . . 9 ⊢ ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
3	2	ralrimiva 3127	. . . . . . . 8 ⊢ (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
4	1	cmpcov2 23336	. . . . . . . 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 594	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
6		elfpw 9256	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin))
7		simplrr 778	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → 𝑐 ∈ Fin)
8		eldifsn 4741	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅))
9		ineq1 4164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (𝑠 ∩ 𝑜) = (𝑥 ∩ 𝑜))
10	9	neeq1d 2990	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → ((𝑠 ∩ 𝑜) ≠ ∅ ↔ (𝑥 ∩ 𝑜) ≠ ∅))
11		simplrl 777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ 𝐶)
12		simplrr 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑥)
13		simprr 773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑜)
14		inelcm 4416	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ 𝑜) → (𝑥 ∩ 𝑜) ≠ ∅)
15	12, 13, 14	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 3647	. . . . . . . . . . . . . . . . . 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 4867	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
18		locfincmp.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑌 = ∪ 𝐶
19	17, 18	eleqtrrdi 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ 𝑌)
20	19	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
21	20	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑌)
22	1, 18	locfinbas 23468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
24	23	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = 𝑌)
25	21, 24	eleqtrrd 2838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
26		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = ∪ 𝑐)
27	25, 26	eleqtrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝑐)
28		eluni2 4866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ∪ 𝑐 ↔ ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
29	27, 28	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
30	16, 29	reximddv 3151	. . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . 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 1935	. . . . . . . . . . . . . . 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 4304	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
34		eliun 4949	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ↔ ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
35	32, 33, 34	3imtr4g 296	. . . . . . . . . . . . . 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 453	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
38	37	ssrdv 3938	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
39		iunfi 9245	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ Fin ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin)
40	39	ex 412	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ Fin → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
41		ssfi 9099	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
42	41	expcom 413	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} → (∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
43	40, 42	sylan9 507	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
44	7, 38, 43	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
45	44	expimpd 453	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
47	46	rexlimdva 3136	. . . . . . 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 15	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∖ {∅}) ∈ Fin)
49		snfi 8982	. . . . . 6 ⊢ {∅} ∈ Fin
50		unfi 9097	. . . . . 6 ⊢ (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
51	48, 49, 50	sylancl 587	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52		ssun1 4129	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {∅})
53		undif1 4427	. . . . . 6 ⊢ ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
54	52, 53	sseqtrri 3982	. . . . 5 ⊢ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55		ssfi 9099	. . . . 5 ⊢ ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
56	51, 54, 55	sylancl 587	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
57	56, 23	jca 511	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
58	57	ex 412	. 2 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59		cmptop 23341	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
60	1, 18	finlocfin 23466	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61	60	3expib 1123	. . 3 ⊢ (𝐽 ∈ Top → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
62	59, 61	syl 17	. 2 ⊢ (𝐽 ∈ Comp → ((𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽)))
63	58, 62	impbid 212	1 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  {crab 3398   ∖ cdif 3897   ∪ cun 3898   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  𝒫 cpw 4553  {csn 4579  ∪ cuni 4862  ∪ ciun 4945  ‘cfv 6491  Fincfn 8885  Topctop 22839  Compccmp 23332  LocFinclocfin 23450

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-en 8886  df-fin 8889  df-top 22840  df-cmp 23333  df-locfin 23453

This theorem is referenced by: (None)