Theorem lfinun 23419

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 23415	. . . . 5 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
2	1	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → 𝐽 ∈ Top)
3		ssequn2 4155	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2730	. . . . . . . . 9 ⊢ ∪ 𝐴 = ∪ 𝐴
8	6, 7	locfinbas 23416	. . . . . . . 8 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴)
9	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ 𝐴)
10	9	uneq1d 4133	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵))
11	5, 10	eqtr3d 2767	. . . . 5 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = (∪ 𝐴 ∪ ∪ 𝐵))
12		uniun 4897	. . . . 5 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
13	11, 12	eqtr4di 2783	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵))
14	6	locfinnei 23417	. . . . . . 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 9221	. . . . . . . . . . . 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 4290	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} = ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
20		unfi 9141	. . . . . . . . . . . 12 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) ∈ Fin)
21	19, 20	eqeltrid 2833	. . . . . . . . . . 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 3149	. . . . . 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 3126	. . . 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 2730	. . 3 ⊢ ∪ (𝐴 ∪ 𝐵) = ∪ (𝐴 ∪ 𝐵)
32	6, 31	islocfin 23411	. 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 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ‘cfv 6514  Fincfn 8921  Topctop 22787  LocFinclocfin 23398

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-om 7846  df-1o 8437  df-en 8922  df-fin 8925  df-top 22788  df-locfin 23401

This theorem is referenced by:  locfinref  33838