Theorem lfinun 23460

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 23456	. . . . 5 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
2	1	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → 𝐽 ∈ Top)
3		ssequn2 4138	. . . . . . . 8 ⊢ (∪ 𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
4	3	biimpi 216	. . . . . . 7 ⊢ (∪ 𝐵 ⊆ ∪ 𝐽 → (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
5	4	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
6		eqid 2733	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2733	. . . . . . . . 9 ⊢ ∪ 𝐴 = ∪ 𝐴
8	6, 7	locfinbas 23457	. . . . . . . 8 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴)
9	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ 𝐴)
10	9	uneq1d 4116	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵))
11	5, 10	eqtr3d 2770	. . . . 5 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = (∪ 𝐴 ∪ ∪ 𝐵))
12		uniun 4883	. . . . 5 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
13	11, 12	eqtr4di 2786	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵))
14	6	locfinnei 23458	. . . . . . 7 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
15	14	ad4ant14 752	. . . . . 6 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
16		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
17		rabfi 9166	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
18	17	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
19		rabun2 4273	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} = ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
20		unfi 9091	. . . . . . . . . . . 12 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) ∈ Fin)
21	19, 20	eqeltrid 2837	. . . . . . . . . . 11 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
22	16, 18, 21	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)
23	22	ex 412	. . . . . . . . 9 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
24	23	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin → {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
25	24	anim2d 612	. . . . . . 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)))
26	25	reximdv 3148	. . . . . 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)))
27	15, 26	mpd 15	. . . . 5 ⊢ ((((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
28	27	ralrimiva 3125	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
29	2, 13, 28	3jca 1128	. . 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)))
30	29	3impa 1109	. 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)))
31		eqid 2733	. . 3 ⊢ ∪ (𝐴 ∪ 𝐵) = ∪ (𝐴 ∪ 𝐵)
32	6, 31	islocfin 23452	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵) ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
33	30, 32	sylibr 234	1 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (𝐴 ∪ 𝐵) ∈ (LocFin‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  {crab 3396   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  ∪ cuni 4860  ‘cfv 6489  Fincfn 8879  Topctop 22828  LocFinclocfin 23439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7806  df-1o 8394  df-en 8880  df-fin 8883  df-top 22829  df-locfin 23442

This theorem is referenced by:  locfinref  33926