MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldval Structured version   Visualization version   GIF version

Theorem cldval 21559
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 21442 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5270 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 5225 . . 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 4838 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2871 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4540 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
97difeq1d 4095 . . . . 5 (𝑗 = 𝐽 → ( 𝑗𝑥) = (𝑋𝑥))
10 eleq12 2899 . . . . 5 ((( 𝑗𝑥) = (𝑋𝑥) ∧ 𝑗 = 𝐽) → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
119, 10mpancom 684 . . . 4 (𝑗 = 𝐽 → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
128, 11rabeqbidv 3483 . . 3 (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
13 df-cld 21555 . . 3 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
1412, 13fvmptg 6759 . 2 ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
155, 14mpdan 683 1 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cdif 3930  𝒫 cpw 4535   cuni 4830  cfv 6348  Topctop 21429  Clsdccld 21552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-cld 21555
This theorem is referenced by:  iscld  21563  mretopd  21628
  Copyright terms: Public domain W3C validator