Theorem locfincmp 23413

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 23410	. . . . . . . . 9 ⊢ ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
3	2	ralrimiva 3125	. . . . . . . 8 ⊢ (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
4	1	cmpcov2 23277	. . . . . . . 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 593	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
6		elfpw 9305	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin))
7		simplrr 777	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → 𝑐 ∈ Fin)
8		eldifsn 4750	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅))
9		ineq1 4176	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (𝑠 ∩ 𝑜) = (𝑥 ∩ 𝑜))
10	9	neeq1d 2984	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → ((𝑠 ∩ 𝑜) ≠ ∅ ↔ (𝑥 ∩ 𝑜) ≠ ∅))
11		simplrl 776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ 𝐶)
12		simplrr 777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑥)
13		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑜)
14		inelcm 4428	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ 𝑜) → (𝑥 ∩ 𝑜) ≠ ∅)
15	12, 13, 14	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 3661	. . . . . . . . . . . . . . . . . 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 4876	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
18		locfincmp.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑌 = ∪ 𝐶
19	17, 18	eleqtrrdi 2839	. . . . . . . . . . . . . . . . . . . . . . 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 23409	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
24	23	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = 𝑌)
25	21, 24	eleqtrrd 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
26		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = ∪ 𝑐)
27	25, 26	eleqtrd 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝑐)
28		eluni2 4875	. . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . . . 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 4316	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
34		eliun 4959	. . . . . . . . . . . . . . 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 3952	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
39		iunfi 9294	. . . . . . . . . . . . 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 9137	. . . . . . . . . . . . 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 584	. . . . . . . . . 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 594	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
47	46	rexlimdva 3134	. . . . . . 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 9014	. . . . . 6 ⊢ {∅} ∈ Fin
50		unfi 9135	. . . . . 6 ⊢ (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
51	48, 49, 50	sylancl 586	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52		ssun1 4141	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {∅})
53		undif1 4439	. . . . . 6 ⊢ ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
54	52, 53	sseqtrri 3996	. . . . 5 ⊢ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55		ssfi 9137	. . . . 5 ⊢ ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
56	51, 54, 55	sylancl 586	. . . 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 23282	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
60	1, 18	finlocfin 23407	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61	60	3expib 1122	. . 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ∪ ciun 4955  ‘cfv 6511  Fincfn 8918  Topctop 22780  Compccmp 23273  LocFinclocfin 23391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-en 8919  df-fin 8922  df-top 22781  df-cmp 23274  df-locfin 23394

This theorem is referenced by: (None)