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Theorem cldval 21107
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 20990 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5014 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 4972 . . 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 4602 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2817 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4320 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
97difeq1d 3889 . . . . 5 (𝑗 = 𝐽 → ( 𝑗𝑥) = (𝑋𝑥))
10 eleq12 2834 . . . . 5 ((( 𝑗𝑥) = (𝑋𝑥) ∧ 𝑗 = 𝐽) → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
119, 10mpancom 679 . . . 4 (𝑗 = 𝐽 → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
128, 11rabeqbidv 3344 . . 3 (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
13 df-cld 21103 . . 3 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
1412, 13fvmptg 6469 . 2 ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
155, 14mpdan 678 1 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cdif 3729  𝒫 cpw 4315   cuni 4594  cfv 6068  Topctop 20977  Clsdccld 21100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-top 20978  df-cld 21103
This theorem is referenced by:  iscld  21111  mretopd  21176
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