Theorem locfincmp 23845

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 23842	. . . . . . . . 9 ⊢ ((𝐶 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
3	2	ralrimiva 3154	. . . . . . . 8 ⊢ (𝐶 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ∧ {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
4	1	cmpcov2 23708	. . . . . . . 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 605	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ∃𝑐 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
6		elfpw 9343	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin))
7		simplrr 790	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → 𝑐 ∈ Fin)
8		eldifsn 4747	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐶 ∖ {∅}) ↔ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ ∅))
9		ineq1 4110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (𝑠 ∩ 𝑜) = (𝑥 ∩ 𝑜))
10	9	neeq1d 3014	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → ((𝑠 ∩ 𝑜) ≠ ∅ ↔ (𝑥 ∩ 𝑜) ≠ ∅))
11		simplrl 789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑥 ∈ 𝐶)
12		simplrr 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑥)
13		simprr 785	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑜 ∈ 𝑐 ∧ 𝑦 ∈ 𝑜)) → 𝑦 ∈ 𝑜)
14		inelcm 4417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ 𝑜) → (𝑥 ∩ 𝑜) ≠ ∅)
15	12, 13, 14	syl2anc 596	. . . . . . . . . . . . . . . . . . 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 3646	. . . . . . . . . . . . . . . . . 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 4871	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ ∪ 𝐶)
18		locfincmp.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑌 = ∪ 𝐶
19	17, 18	eleqtrrdi 2871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐶) → 𝑦 ∈ 𝑌)
20	19	ancoms 464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
21	20	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑌)
22	1, 18	locfinbas 23841	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐶 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
23	22	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑌)
24	23	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = 𝑌)
25	21, 24	eleqtrrd 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋)
26		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑋 = ∪ 𝑐)
27	25, 26	eleqtrd 2862	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝑐)
28		eluni2 4870	. . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . 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 1965	. . . . . . . . . . . . . . 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 4299	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
34		eliun 4954	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . 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 3914	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅})
39		iunfi 9332	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ Fin ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin)
40	39	ex 418	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ Fin → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin))
41		ssfi 9188	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (𝐶 ∖ {∅}) ∈ Fin)
42	41	expcom 419	. . . . . . . . . . . 12 ⊢ ((𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} → (∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
43	40, 42	sylan9 517	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ Fin ∧ (𝐶 ∖ {∅}) ⊆ ∪ 𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅}) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
44	7, 38, 43	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ (𝑐 ⊆ 𝐽 ∧ 𝑐 ∈ Fin)) ∧ 𝑋 = ∪ 𝑐) → (∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin → (𝐶 ∖ {∅}) ∈ Fin))
45	44	expimpd 459	. . . . . . . . 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 606	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) ∧ 𝑐 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑐 ∧ ∀𝑜 ∈ 𝑐 {𝑠 ∈ 𝐶 ∣ (𝑠 ∩ 𝑜) ≠ ∅} ∈ Fin) → (𝐶 ∖ {∅}) ∈ Fin))
47	46	rexlimdva 3163	. . . . . . 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 9071	. . . . . 6 ⊢ {∅} ∈ Fin
50		unfi 9186	. . . . . 6 ⊢ (((𝐶 ∖ {∅}) ∈ Fin ∧ {∅} ∈ Fin) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
51	48, 49, 50	sylancl 598	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → ((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin)
52		ssun1 4176	. . . . . 6 ⊢ 𝐶 ⊆ (𝐶 ∪ {∅})
53		undif1 4429	. . . . . 6 ⊢ ((𝐶 ∖ {∅}) ∪ {∅}) = (𝐶 ∪ {∅})
54	52, 53	sseqtrri 3957	. . . . 5 ⊢ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})
55		ssfi 9188	. . . . 5 ⊢ ((((𝐶 ∖ {∅}) ∪ {∅}) ∈ Fin ∧ 𝐶 ⊆ ((𝐶 ∖ {∅}) ∪ {∅})) → 𝐶 ∈ Fin)
56	51, 54, 55	sylancl 598	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → 𝐶 ∈ Fin)
57	56, 23	jca 521	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝐶 ∈ (LocFin‘𝐽)) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌))
58	57	ex 418	. 2 ⊢ (𝐽 ∈ Comp → (𝐶 ∈ (LocFin‘𝐽) → (𝐶 ∈ Fin ∧ 𝑋 = 𝑌)))
59		cmptop 23713	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
60	1, 18	finlocfin 23839	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐶 ∈ (LocFin‘𝐽))
61	60	3expib 1140	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866  ∪ ciun 4950  ‘cfv 6538  Fincfn 8973  Topctop 23211  Compccmp 23704  LocFinclocfin 23823

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-en 8974  df-fin 8977  df-top 23212  df-cmp 23705  df-locfin 23826

This theorem is used by: (None)