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Theorem cldval 12050
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 11957 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4044 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 4011 . . 3 (𝒫 𝑋 ∈ V → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V)
52, 3, 43syl 17 . 2 (𝐽 ∈ Top → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V)
6 unieq 3692 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2150 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 3462 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
97difeq1d 3140 . . . . 5 (𝑗 = 𝐽 → ( 𝑗𝑥) = (𝑋𝑥))
10 eleq12 2164 . . . . 5 ((( 𝑗𝑥) = (𝑋𝑥) ∧ 𝑗 = 𝐽) → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
119, 10mpancom 416 . . . 4 (𝑗 = 𝐽 → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
128, 11rabeqbidv 2636 . . 3 (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
13 df-cld 12046 . . 3 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
1412, 13fvmptg 5429 . 2 ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
155, 14mpdan 415 1 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1299  wcel 1448  {crab 2379  Vcvv 2641  cdif 3018  𝒫 cpw 3457   cuni 3683  cfv 5059  Topctop 11946  Clsdccld 12043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-top 11947  df-cld 12046
This theorem is referenced by:  iscld  12054
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