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Theorem iscld 22394
Description: The predicate "the class 𝑆 is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iscld (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))

Proof of Theorem iscld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = 𝐽
21cldval 22390 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
32eleq2d 2820 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽}))
4 difeq2 4077 . . . . 5 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
54eleq1d 2819 . . . 4 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐽 ↔ (𝑋𝑆) ∈ 𝐽))
65elrab 3646 . . 3 (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽))
73, 6bitrdi 287 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
81topopn 22271 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
9 elpw2g 5302 . . . 4 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
108, 9syl 17 . . 3 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1110anbi1d 631 . 2 (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
127, 11bitrd 279 1 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3406  cdif 3908  wss 3911  𝒫 cpw 4561   cuni 4866  cfv 6497  Topctop 22258  Clsdccld 22383
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
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-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-cld 22386
This theorem is referenced by:  iscld2  22395  cldss  22396  cldopn  22398  topcld  22402  discld  22456  indiscld  22458  restcld  22539  ordtcld1  22564  ordtcld2  22565  hauscmp  22774  txcld  22970  ptcld  22980  qtopcld  23080  opnsubg  23475  sszcld  24196  ist0cld  32471  stoweidlem57  44384
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