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Theorem iuncld 12316
 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 3832 . . 3 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
21eleq1d 2209 . 2 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3 iuneq1 3832 . . 3 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
43eleq1d 2209 . 2 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)))
5 iuneq1 3832 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
65eleq1d 2209 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7 iuneq1 3832 . . 3 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
87eleq1d 2209 . 2 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)))
9 0iun 3876 . . . 4 𝑥 ∈ ∅ 𝐵 = ∅
10 0cld 12313 . . . 4 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
119, 10eqeltrid 2227 . . 3 (𝐽 ∈ Top → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12113ad2ant1 1003 . 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 3038 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
15 csbeq1a 3015 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1614, 15iunxsngf 3896 . . . . . . 7 (𝑧 ∈ V → 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵)
1716elv 2693 . . . . . 6 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
18 simprr 522 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
1918eldifad 3085 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
20 simpll3 1023 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
2114nfel1 2293 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)
2215eleq1d 2209 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)))
2321, 22rspc 2786 . . . . . . 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, 24eqeltrid 2227 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
2625adantr 274 . . . 4 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
27 iunxun 3898 . . . . 5 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
28 uncld 12314 . . . . 5 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
2927, 28eqeltrid 2227 . . . 4 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
3013, 26, 29syl2anc 409 . . 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 983 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
332, 4, 6, 8, 12, 31, 32findcard2sd 6792 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 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  Vcvv 2689  ⦋csb 3006   ∖ cdif 3071   ∪ cun 3072   ⊆ wss 3074  ∅c0 3366  {csn 3530  ∪ cuni 3742  ∪ ciun 3819  ‘cfv 5129  Fincfn 6640  Topctop 12196  Clsdccld 12293 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-iord 4294  df-on 4296  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-er 6435  df-en 6641  df-fin 6643  df-top 12197  df-cld 12296 This theorem is referenced by:  unicld  12317
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