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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 ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 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.
StepHypRef Expression
1 iuneq1 3773 . . 3 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
21eleq1d 2168 . 2 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3 iuneq1 3773 . . 3 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
43eleq1d 2168 . 2 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)))
5 iuneq1 3773 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
65eleq1d 2168 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7 iuneq1 3773 . . 3 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
87eleq1d 2168 . 2 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)))
9 0iun 3817 . . . 4 𝑥 ∈ ∅ 𝐵 = ∅
10 0cld 12063 . . . 4 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
119, 10syl5eqel 2186 . . 3 (𝐽 ∈ Top → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12113ad2ant1 970 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
13 simpr 109 . . . 4 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽))
14 nfcsb1v 2985 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
15 csbeq1a 2963 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1614, 15iunxsngf 3837 . . . . . . 7 (𝑧 ∈ V → 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵)
1716elv 2645 . . . . . 6 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
18 simprr 502 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
1918eldifad 3032 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
20 simpll3 990 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
2114nfel1 2251 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)
2215eleq1d 2168 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)))
2321, 22rspc 2738 . . . . . . 7 (𝑧𝐴 → (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)))
2419, 20, 23sylc 62 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽))
2517, 24syl5eqel 2186 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
2625adantr 272 . . . 4 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
27 iunxun 3839 . . . . 5 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
28 uncld 12064 . . . . 5 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
2927, 28syl5eqel 2186 . . . 4 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
3013, 26, 29syl2anc 406 . . 3 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
3130ex 114 . 2 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
32 simp2 950 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
332, 4, 6, 8, 12, 31, 32findcard2sd 6715 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff set class
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
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