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Theorem iincld 23063
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 2735 . . . . . . . 8 𝐽 = 𝐽
21cldss 23053 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
3 dfss4 4275 . . . . . . 7 (𝐵 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
42, 3sylib 218 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
54ralimi 3081 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
6 iineq2 5017 . . . . 5 (∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
75, 6syl 17 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
87adantl 481 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
9 iindif2 5082 . . . 4 (𝐴 ≠ ∅ → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
109adantr 480 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
118, 10eqtr3d 2777 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
12 r19.2z 4501 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
13 cldrcl 23050 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1413rexlimivw 3149 . . . 4 (∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1512, 14syl 17 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
161cldopn 23055 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
1716ralimi 3081 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
1817adantl 481 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
19 iunopn 22920 . . . 4 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
2015, 18, 19syl2anc 584 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
211opncld 23057 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2215, 20, 21syl2anc 584 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2311, 22eqeltrd 2839 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cdif 3960  wss 3963  c0 4339   cuni 4912   ciun 4996   ciin 4997  cfv 6563  Topctop 22915  Clsdccld 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-top 22916  df-cld 23043
This theorem is referenced by:  intcld  23064  riincld  23068  hauscmplem  23430  ubthlem1  30899
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