Theorem lfinun 23713

Description: Adding a finite set preserves locally finite covers. (Contributed by Thierry Arnoux, 31-Jan-2020.)

Assertion

Ref	Expression
lfinun	⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽))

Proof of Theorem lfinun

Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		locfintop 23709	. . . . 5 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
2	1	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → 𝐽 ∈ Top)
3		ssequn2 4142	. . . . . . 7 ⊢ (∪ 𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
4	3	bilani 510	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
5		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝐴 = ∪ 𝐴
7	5, 6	locfinbas 23710	. . . . . . . 8 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴)
8	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ 𝐴)
9	8	uneq1d 4121	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵))
10	4, 9	eqtr3d 2802	. . . . 5 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = (∪ 𝐴 ∪ ∪ 𝐵))
11		uniun 4897	. . . . 5 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
12	10, 11	eqtr4di 2818	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵))
13	5	locfinnei 23711	. . . . . . 7 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
14	13	ad4ant14 765	. . . . . 6 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
15		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
16		rabfi 9234	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
17	16	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
18		rabun2 4277	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} = ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
19		unfi 9158	. . . . . . . . . . . 12 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) ∈ Fin)
20	18, 19	eqeltrid 2869	. . . . . . . . . . 11 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
21	15, 17, 20	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
22	21	ex 418	. . . . . . . . 9 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
23	22	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
24	23	anim2d 624	. . . . . . 7 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
25	24	reximdv 3182	. . . . . 6 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
26	14, 25	mpd 16	. . . . 5 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
27	26	ralrimiva 3159	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
28	2, 12, 27	3jca 1146	. . 3 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
29	28	3impa 1127	. 2 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
30		eqid 2765	. . 3 ⊢ ∪ (𝐴 ∪ 𝐵) = ∪ (𝐴 ∪ 𝐵)
31	5, 30	islocfin 23705	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
32	29, 31	sylibr 237	1 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  ∪ cuni 4874  ‘cfv 6540  Fincfn 8945  Topctop 23080  LocFinclocfin 23692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7865  df-1o 8455  df-en 8946  df-fin 8949  df-top 23081  df-locfin 23695

This theorem is used by:  locfinref  34271