Theorem iuncld 14657

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 3945	. . 3 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
2	1	eleq1d 2275	. 2 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3		iuneq1 3945	. . 3 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
4	3	eleq1d 2275	. 2 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽)))
5		iuneq1 3945	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
6	5	eleq1d 2275	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7		iuneq1 3945	. . 3 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
8	7	eleq1d 2275	. 2 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)))
9		0iun 3990	. . . 4 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
10		0cld 14654	. . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
11	9, 10	eqeltrid 2293	. . 3 ⊢ (𝐽 ∈ Top → ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12	11	3ad2ant1 1021	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
13		simpr 110	. . . 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 3130	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15		csbeq1a 3106	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
16	14, 15	iunxsngf 4010	. . . . . . 7 ⊢ (𝑧 ∈ V → ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
17	16	elv 2777	. . . . . 6 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
18		simprr 531	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
19	18	eldifad 3181	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
20		simpll3 1041	. . . . . . 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 2360	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)
22	15	eleq1d 2275	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ (Clsd‘𝐽)))
23	21, 22	rspc 2875	. . . . . . 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 2293	. . . . 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 276	. . . 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 4012	. . . . 5 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
28		uncld 14655	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
29	27, 28	eqeltrid 2293	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
30	13, 26, 29	syl2anc 411	. . 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 115	. 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 1001	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
33	2, 4, 6, 8, 12, 31, 32	findcard2sd 7003	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ⦋csb 3097   ∖ cdif 3167   ∪ cun 3168   ⊆ wss 3170  ∅c0 3464  {csn 3637  ∪ cuni 3855  ∪ ciun 3932  ‘cfv 5279  Fincfn 6839  Topctop 14539  Clsdccld 14634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-er 6632  df-en 6840  df-fin 6842  df-top 14540  df-cld 14637

This theorem is referenced by:  unicld  14658