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Theorem iscld 20879
Description: The predicate "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 20875 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
32eleq2d 2716 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽}))
4 difeq2 3755 . . . . 5 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
54eleq1d 2715 . . . 4 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐽 ↔ (𝑋𝑆) ∈ 𝐽))
65elrab 3396 . . 3 (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽))
73, 6syl6bb 276 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
81topopn 20759 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
9 elpw2g 4857 . . . 4 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
108, 9syl 17 . . 3 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1110anbi1d 741 . 2 (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
127, 11bitrd 268 1 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  cdif 3604  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-cld 20871
This theorem is referenced by:  iscld2  20880  cldss  20881  cldopn  20883  topcld  20887  discld  20941  indiscld  20943  restcld  21024  ordtcld1  21049  ordtcld2  21050  hauscmp  21258  txcld  21454  ptcld  21464  qtopcld  21564  opnsubg  21958  sszcld  22667  stoweidlem57  40592
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