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Theorem iincld 21654
 Description: The indexed intersection of a collection 𝐵(𝑥) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
iincld ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iincld
StepHypRef Expression
1 eqid 2798 . . . . . . . 8 𝐽 = 𝐽
21cldss 21644 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
3 dfss4 4185 . . . . . . 7 (𝐵 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
42, 3sylib 221 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
54ralimi 3128 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
6 iineq2 4902 . . . . 5 (∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
75, 6syl 17 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
87adantl 485 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
9 iindif2 4963 . . . 4 (𝐴 ≠ ∅ → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
109adantr 484 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
118, 10eqtr3d 2835 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
12 r19.2z 4398 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
13 cldrcl 21641 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1413rexlimivw 3241 . . . 4 (∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1512, 14syl 17 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
161cldopn 21646 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
1716ralimi 3128 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
1817adantl 485 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
19 iunopn 21513 . . . 4 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
2015, 18, 19syl2anc 587 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
211opncld 21648 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2215, 20, 21syl2anc 587 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2311, 22eqeltrd 2890 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4801  ∪ ciun 4882  ∩ ciin 4883  ‘cfv 6325  Topctop 21508  Clsdccld 21631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333  df-top 21509  df-cld 21634 This theorem is referenced by:  intcld  21655  riincld  21659  hauscmplem  22021  ubthlem1  28663
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