Theorem lfinun 23899

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 23895	. . . . 5 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
2	1	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → 𝐽 ∈ Top)
3		ssequn2 4190	. . . . . . 7 ⊢ (∪ 𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
4	3	bilani 510	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
5		eqid 2763	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2763	. . . . . . . . 9 ⊢ ∪ 𝐴 = ∪ 𝐴
7	5, 6	locfinbas 23896	. . . . . . . 8 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴)
8	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ 𝐴)
9	8	uneq1d 4169	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵))
10	4, 9	eqtr3d 2800	. . . . 5 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = (∪ 𝐴 ∪ ∪ 𝐵))
11		uniun 4895	. . . . 5 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
12	10, 11	eqtr4di 2816	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵))
13	5	locfinnei 23897	. . . . . . 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 9300	. . . . . . . . . . . 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 4272	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} = ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
19		unfi 9224	. . . . . . . . . . . 12 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) ∈ Fin)
20	18, 19	eqeltrid 2868	. . . . . . . . . . 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 3181	. . . . . 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 3158	. . . 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 2763	. . 3 ⊢ ∪ (𝐴 ∪ 𝐵) = ∪ (𝐴 ∪ 𝐵)
31	5, 30	islocfin 23891	. 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ∪ cuni 4872  ‘cfv 6566  Fincfn 9011  Topctop 23266  LocFinclocfin 23878

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1o 8514  df-en 9012  df-fin 9015  df-top 23267  df-locfin 23881

This theorem is used by:  locfinref  34584