Theorem iuncld 12755

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 3879	. . 3 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
2	1	eleq1d 2235	. 2 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3		iuneq1 3879	. . 3 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
4	3	eleq1d 2235	. 2 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)))
5		iuneq1 3879	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
6	5	eleq1d 2235	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7		iuneq1 3879	. . 3 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
8	7	eleq1d 2235	. 2 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)))
9		0iun 3923	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
10		0cld 12752	. . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
11	9, 10	eqeltrid 2253	. . 3 ⊢ (𝐽 ∈ Top → ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12	11	3ad2ant1 1008	. 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 3078	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15		csbeq1a 3054	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
16	14, 15	iunxsngf 3943	. . . . . . 7 ⊢ (𝑧 ∈ V → ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
17	16	elv 2730	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
18		simprr 522	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
19	18	eldifad 3127	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
20		simpll3 1028	. . . . . . 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 2319	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)
22	15	eleq1d 2235	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)))
23	21, 22	rspc 2824	. . . . . . 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	eqeltrid 2253	. . . . 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 274	. . . 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 3945	. . . . 5 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
28		uncld 12753	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
29	27, 28	eqeltrid 2253	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
30	13, 26, 29	syl2anc 409	. . 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 988	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
33	2, 4, 6, 8, 12, 31, 32	findcard2sd 6858	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444  Vcvv 2726  ⦋csb 3045   ∖ cdif 3113   ∪ cun 3114   ⊆ wss 3116  ∅c0 3409  {csn 3576  ∪ cuni 3789  ∪ ciun 3866  ‘cfv 5188  Fincfn 6706  Topctop 12635  Clsdccld 12732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707  df-fin 6709  df-top 12636  df-cld 12735

This theorem is referenced by:  unicld  12756