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Theorem iuncld 14092
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 3914 . . 3 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
21eleq1d 2258 . 2 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3 iuneq1 3914 . . 3 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
43eleq1d 2258 . 2 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)))
5 iuneq1 3914 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
65eleq1d 2258 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7 iuneq1 3914 . . 3 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
87eleq1d 2258 . 2 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)))
9 0iun 3959 . . . 4 𝑥 ∈ ∅ 𝐵 = ∅
10 0cld 14089 . . . 4 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
119, 10eqeltrid 2276 . . 3 (𝐽 ∈ Top → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12113ad2ant1 1020 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
13 simpr 110 . . . 4 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽))
14 nfcsb1v 3105 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
15 csbeq1a 3081 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1614, 15iunxsngf 3979 . . . . . . 7 (𝑧 ∈ V → 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵)
1716elv 2756 . . . . . 6 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵
18 simprr 531 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧 ∈ (𝐴𝑦))
1918eldifad 3155 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
20 simpll3 1040 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
2114nfel1 2343 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)
2215eleq1d 2258 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)))
2321, 22rspc 2850 . . . . . . 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 2276 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
2625adantr 276 . . . 4 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽))
27 iunxun 3981 . . . . 5 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
28 uncld 14090 . . . . 5 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
2927, 28eqeltrid 2276 . . . 4 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
3013, 26, 29syl2anc 411 . . 3 (((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) ∧ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽))
3130ex 115 . 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 1000 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
332, 4, 6, 8, 12, 31, 32findcard2sd 6921 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  csb 3072  cdif 3141  cun 3142  wss 3144  c0 3437  {csn 3607   cuni 3824   ciun 3901  cfv 5235  Fincfn 6767  Topctop 13974  Clsdccld 14069
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6560  df-en 6768  df-fin 6770  df-top 13975  df-cld 14072
This theorem is referenced by:  unicld  14093
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