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Theorem iuncld 13618
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 3900 . . 3 (𝑤 = ∅ → 𝑥𝑤 𝐵 = 𝑥 ∈ ∅ 𝐵)
21eleq1d 2246 . 2 (𝑤 = ∅ → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽)))
3 iuneq1 3900 . . 3 (𝑤 = 𝑦 𝑥𝑤 𝐵 = 𝑥𝑦 𝐵)
43eleq1d 2246 . 2 (𝑤 = 𝑦 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽)))
5 iuneq1 3900 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑥𝑤 𝐵 = 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
65eleq1d 2246 . 2 (𝑤 = (𝑦 ∪ {𝑧}) → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ (Clsd‘𝐽)))
7 iuneq1 3900 . . 3 (𝑤 = 𝐴 𝑥𝑤 𝐵 = 𝑥𝐴 𝐵)
87eleq1d 2246 . 2 (𝑤 = 𝐴 → ( 𝑥𝑤 𝐵 ∈ (Clsd‘𝐽) ↔ 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)))
9 0iun 3945 . . . 4 𝑥 ∈ ∅ 𝐵 = ∅
10 0cld 13615 . . . 4 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
119, 10eqeltrid 2264 . . 3 (𝐽 ∈ Top → 𝑥 ∈ ∅ 𝐵 ∈ (Clsd‘𝐽))
12113ad2ant1 1018 . 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 3091 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
15 csbeq1a 3067 . . . . . . . 8 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
1614, 15iunxsngf 3965 . . . . . . 7 (𝑧 ∈ V → 𝑥 ∈ {𝑧}𝐵 = 𝑧 / 𝑥𝐵)
1716elv 2742 . . . . . 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 3141 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → 𝑧𝐴)
20 simpll3 1038 . . . . . . 7 ((((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) ∧ 𝑦 ∈ Fin) ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
2114nfel1 2330 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)
2215eleq1d 2246 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵 ∈ (Clsd‘𝐽) ↔ 𝑧 / 𝑥𝐵 ∈ (Clsd‘𝐽)))
2321, 22rspc 2836 . . . . . . 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 2264 . . . . 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 3967 . . . . 5 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵)
28 uncld 13616 . . . . 5 (( 𝑥𝑦 𝐵 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ {𝑧}𝐵 ∈ (Clsd‘𝐽)) → ( 𝑥𝑦 𝐵 𝑥 ∈ {𝑧}𝐵) ∈ (Clsd‘𝐽))
2927, 28eqeltrid 2264 . . . 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 998 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ Fin)
332, 4, 6, 8, 12, 31, 32findcard2sd 6892 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 978   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  csb 3058  cdif 3127  cun 3128  wss 3130  c0 3423  {csn 3593   cuni 3810   ciun 3887  cfv 5217  Fincfn 6740  Topctop 13500  Clsdccld 13595
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-er 6535  df-en 6741  df-fin 6743  df-top 13501  df-cld 13598
This theorem is referenced by:  unicld  13619
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