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Theorem cldval 23031
Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1 𝑋 = 𝐽
Assertion
Ref Expression
cldval (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem cldval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4 𝑋 = 𝐽
21topopn 22912 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5378 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 5337 . . 3 (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V)
6 unieq 4918 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2795 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4617 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
97difeq1d 4125 . . . . 5 (𝑗 = 𝐽 → ( 𝑗𝑥) = (𝑋𝑥))
10 eleq12 2831 . . . . 5 ((( 𝑗𝑥) = (𝑋𝑥) ∧ 𝑗 = 𝐽) → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
119, 10mpancom 688 . . . 4 (𝑗 = 𝐽 → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
128, 11rabeqbidv 3455 . . 3 (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
13 df-cld 23027 . . 3 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
1412, 13fvmptg 7014 . 2 ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
155, 14mpdan 687 1 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cdif 3948  𝒫 cpw 4600   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by:  iscld  23035  mretopd  23100
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