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Theorem iscld 12331
 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 12327 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
32eleq2d 2210 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽}))
4 difeq2 3194 . . . . 5 (𝑥 = 𝑆 → (𝑋𝑥) = (𝑋𝑆))
54eleq1d 2209 . . . 4 (𝑥 = 𝑆 → ((𝑋𝑥) ∈ 𝐽 ↔ (𝑋𝑆) ∈ 𝐽))
65elrab 2845 . . 3 (𝑆 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽))
73, 6syl6bb 195 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
81topopn 12234 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
9 elpw2g 4090 . . . 4 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
108, 9syl 14 . . 3 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1110anbi1d 461 . 2 (𝐽 ∈ Top → ((𝑆 ∈ 𝒫 𝑋 ∧ (𝑋𝑆) ∈ 𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
127, 11bitrd 187 1 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {crab 2421   ∖ cdif 3074   ⊆ wss 3077  𝒫 cpw 3516  ∪ cuni 3745  ‘cfv 5132  Topctop 12223  Clsdccld 12320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-iota 5097  df-fun 5134  df-fv 5140  df-top 12224  df-cld 12323 This theorem is referenced by:  iscld2  12332  cldss  12333  cldopn  12335  topcld  12337  discld  12364
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