Theorem lfinun 23837

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 23833	. . . . 5 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
2	1	ad2antrr 739	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → 𝐽 ∈ Top)
3		ssequn2 4135	. . . . . . 7 ⊢ (∪ 𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
4	3	bilani 510	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = ∪ 𝐽)
5		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
6		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐴 = ∪ 𝐴
7	5, 6	locfinbas 23834	. . . . . . . 8 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∪ 𝐽 = ∪ 𝐴)
8	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ 𝐴)
9	8	uneq1d 4114	. . . . . 6 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → (∪ 𝐽 ∪ ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵))
10	4, 9	eqtr3d 2798	. . . . 5 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = (∪ 𝐴 ∪ ∪ 𝐵))
11		uniun 4890	. . . . 5 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)
12	10, 11	eqtr4di 2814	. . . 4 ⊢ (((𝐴 ∈ (LocFin‘𝐽) ∧ 𝐵 ∈ Fin) ∧ ∪ 𝐵 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ (𝐴 ∪ 𝐵))
13	5	locfinnei 23835	. . . . . . 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 9255	. . . . . . . . . . . 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 4270	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (𝐴 ∪ 𝐵) ∣ (𝑠 ∩ 𝑛) ≠ ∅} = ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅})
19		unfi 9179	. . . . . . . . . . . 12 ⊢ (({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ∧ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∪ {𝑠 ∈ 𝐵 ∣ (𝑠 ∩ 𝑛) ≠ ∅}) ∈ Fin)
20	18, 19	eqeltrid 2865	. . . . . . . . . . 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 3178	. . . . . 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 3155	. . . 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 2761	. . 3 ⊢ ∪ (𝐴 ∪ 𝐵) = ∪ (𝐴 ∪ 𝐵)
31	5, 30	islocfin 23829	. 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 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ‘cfv 6537  Fincfn 8966  Topctop 23204  LocFinclocfin 23816

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7876  df-1o 8469  df-en 8967  df-fin 8970  df-top 23205  df-locfin 23819

This theorem is used by:  locfinref  34466