Theorem locfincmp 22585

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 22582	. . . . . . . . 9 ⊢ ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
3	2	ralrimiva 3107	. . . . . . . 8 ⊢ (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
4	1	cmpcov2 22449	. . . . . . . 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 592	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
6		elfpw 9051	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin))
7		simplrr 774	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → 𝑐 ∈ Fin)
8		eldifsn 4717	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅))
9		ineq1 4136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (𝑠 ∩ 𝑜) = (𝑥 ∩ 𝑜))
10	9	neeq1d 3002	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → ((𝑠 ∩ 𝑜) ≠ ∅ ↔ (𝑥 ∩ 𝑜) ≠ ∅))
11		simplrl 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ 𝐶)
12		simplrr 774	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑥)
13		simprr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑜)
14		inelcm 4395	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ 𝑜) → (𝑥 ∩ 𝑜) ≠ ∅)
15	12, 13, 14	syl2anc 583	. . . . . . . . . . . . . . . . . . 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 3619	. . . . . . . . . . . . . . . . . 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 4841	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
18		locfincmp.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑌 = ∪ 𝐶
19	17, 18	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . . . . . 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 22581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
24	23	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = 𝑌)
25	21, 24	eleqtrrd 2842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
26		simplr 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = ∪ 𝑐)
27	25, 26	eleqtrd 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝑐)
28		eluni2 4840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ∪ 𝑐 ↔ ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
29	27, 28	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → ∃𝑜 ∈ 𝑐 𝑦 ∈ 𝑜)
30	16, 29	reximddv 3203	. . . . . . . . . . . . . . . . 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 1937	. . . . . . . . . . . . . . 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 4277	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
34		eliun 4925	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ↔ ∃𝑜 ∈ 𝑐 𝑥 ∈ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
35	32, 33, 34	3imtr4g 295	. . . . . . . . . . . . . 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	syl5bi 241	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝑥 ∈ (𝐶 ∖ {∅}) → 𝑥 ∈ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}))
38	37	ssrdv 3923	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
39		iunfi 9037	. . . . . . . . . . . . 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 8918	. . . . . . . . . . . . 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 583	. . . . . . . . . 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 593	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
47	46	rexlimdva 3212	. . . . . . 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 8788	. . . . . 6 ⊢ {∅} ∈ Fin
50		unfi 8917	. . . . . 6 ⊢ (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
51	48, 49, 50	sylancl 585	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52		ssun1 4102	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {∅})
53		undif1 4406	. . . . . 6 ⊢ ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
54	52, 53	sseqtrri 3954	. . . . 5 ⊢ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55		ssfi 8918	. . . . 5 ⊢ ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
56	51, 54, 55	sylancl 585	. . . 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 22454	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
60	1, 18	finlocfin 22579	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61	60	3expib 1120	. . 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 211	1 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) ↔ (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836  ∪ ciun 4921  ‘cfv 6418  Fincfn 8691  Topctop 21950  Compccmp 22445  LocFinclocfin 22563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695  df-top 21951  df-cmp 22446  df-locfin 22566

This theorem is referenced by: (None)