Theorem iuncld 12066

Description: A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)

Hypothesis

Ref	Expression
iuncld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iuncld	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝐽   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑋(𝑥)

Proof of Theorem iuncld

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 3773	. . 3 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
2	1	eleq1d 2168	. 2 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3		iuneq1 3773	. . 3 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
4	3	eleq1d 2168	. 2 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)))
5		iuneq1 3773	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
6	5	eleq1d 2168	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7		iuneq1 3773	. . 3 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
8	7	eleq1d 2168	. 2 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)))
9		0iun 3817	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
10		0cld 12063	. . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
11	9, 10	syl5eqel 2186	. . 3 ⊢ (𝐽 ∈ Top → ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12	11	3ad2ant1 970	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
13		simpr 109	. . . 4 ⊢ (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽))
14		nfcsb1v 2985	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15		csbeq1a 2963	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
16	14, 15	iunxsngf 3837	. . . . . . 7 ⊢ (𝑧 ∈ V → ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
17	16	elv 2645	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
18		simprr 502	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
19	18	eldifad 3032	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
20		simpll3 990	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
21	14	nfel1 2251	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)
22	15	eleq1d 2168	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)))
23	21, 22	rspc 2738	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)))
24	19, 20, 23	sylc 62	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽))
25	17, 24	syl5eqel 2186	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
26	25	adantr 272	. . . 4 ⊢ (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
27		iunxun 3839	. . . . 5 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
28		uncld 12064	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
29	27, 28	syl5eqel 2186	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
30	13, 26, 29	syl2anc 406	. . 3 ⊢ (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
31	30	ex 114	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
32		simp2 950	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
33	2, 4, 6, 8, 12, 31, 32	findcard2sd 6715	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  ∀wral 2375  Vcvv 2641  ⦋csb 2955   ∖ cdif 3018   ∪ cun 3019   ⊆ wss 3021  ∅c0 3310  {csn 3474  ∪ cuni 3683  ∪ ciun 3760  ‘cfv 5059  Fincfn 6564  Topctop 11946  Clsdccld 12043

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-fin 6567  df-top 11947  df-cld 12046

This theorem is referenced by:  unicld  12067