MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iincld Structured version   Visualization version   GIF version

Theorem iincld 20653
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 2610 . . . . . . . 8 𝐽 = 𝐽
21cldss 20643 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
3 dfss4 3820 . . . . . . 7 (𝐵 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
42, 3sylib 207 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
54ralimi 2936 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
6 iineq2 4474 . . . . 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 4525 . . . 4 (𝐴 ≠ ∅ → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
109adantr 480 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
118, 10eqtr3d 2646 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
12 r19.2z 4012 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
13 cldrcl 20640 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1413rexlimivw 3011 . . . 4 (∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1512, 14syl 17 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
161cldopn 20645 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
1716ralimi 2936 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
1817adantl 481 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
19 iunopn 20528 . . . 4 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
2015, 18, 19syl2anc 691 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
211opncld 20647 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2215, 20, 21syl2anc 691 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2311, 22eqeltrd 2688 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  cdif 3537  wss 3540  c0 3874   cuni 4372   ciun 4455   ciin 4456  cfv 5804  Topctop 20517  Clsdccld 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-top 20521  df-cld 20633
This theorem is referenced by:  intcld  20654  riincld  20658  hauscmplem  21019  ubthlem1  27110
  Copyright terms: Public domain W3C validator