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Theorem iincld 22543
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 2733 . . . . . . . 8 𝐽 = 𝐽
21cldss 22533 . . . . . . 7 (𝐵 ∈ (Clsd‘𝐽) → 𝐵 𝐽)
3 dfss4 4259 . . . . . . 7 (𝐵 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
42, 3sylib 217 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
54ralimi 3084 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵)
6 iineq2 5018 . . . . 5 (∀𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝐵 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
75, 6syl 17 . . . 4 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
87adantl 483 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = 𝑥𝐴 𝐵)
9 iindif2 5081 . . . 4 (𝐴 ≠ ∅ → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
109adantr 482 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽 ∖ ( 𝐽𝐵)) = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
118, 10eqtr3d 2775 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 = ( 𝐽 𝑥𝐴 ( 𝐽𝐵)))
12 r19.2z 4495 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
13 cldrcl 22530 . . . . 5 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1413rexlimivw 3152 . . . 4 (∃𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1512, 14syl 17 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
161cldopn 22535 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → ( 𝐽𝐵) ∈ 𝐽)
1716ralimi 3084 . . . . 5 (∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
1817adantl 483 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
19 iunopn 22400 . . . 4 ((𝐽 ∈ Top ∧ ∀𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
2015, 18, 19syl2anc 585 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽)
211opncld 22537 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝐴 ( 𝐽𝐵) ∈ 𝐽) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2215, 20, 21syl2anc 585 . 2 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → ( 𝐽 𝑥𝐴 ( 𝐽𝐵)) ∈ (Clsd‘𝐽))
2311, 22eqeltrd 2834 1 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝑥𝐴 𝐵 ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  cdif 3946  wss 3949  c0 4323   cuni 4909   ciun 4998   ciin 4999  cfv 6544  Topctop 22395  Clsdccld 22520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-top 22396  df-cld 22523
This theorem is referenced by:  intcld  22544  riincld  22548  hauscmplem  22910  ubthlem1  30123
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