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Theorem iincld 21065
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 2760 . . . . . . . 8 𝐽 = 𝐽
21cldss 21055 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
3 dfss4 4001 . . . . . . 7 (𝐵 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
42, 3sylib 208 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
54ralimi 3090 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
6 iineq2 4690 . . . . 5 (∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
75, 6syl 17 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
87adantl 473 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
9 iindif2 4741 . . . 4 (𝐴 ≠ ∅ → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
109adantr 472 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
118, 10eqtr3d 2796 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
12 r19.2z 4204 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
13 cldrcl 21052 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1413rexlimivw 3167 . . . 4 (∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1512, 14syl 17 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
161cldopn 21057 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
1716ralimi 3090 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
1817adantl 473 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
19 iunopn 20925 . . . 4 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
2015, 18, 19syl2anc 696 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
211opncld 21059 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2215, 20, 21syl2anc 696 . 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 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  cdif 3712  wss 3715  c0 4058   cuni 4588   ciun 4672   ciin 4673  cfv 6049  Topctop 20920  Clsdccld 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20921  df-cld 21045
This theorem is referenced by:  intcld  21066  riincld  21070  hauscmplem  21431  ubthlem1  28056
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