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Theorem cldval 13232
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 13139 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4177 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 4143 . . 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 3816 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2228 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 3579 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
97difeq1d 3252 . . . . 5 (𝑗 = 𝐽 → ( 𝑗𝑥) = (𝑋𝑥))
10 eleq12 2242 . . . . 5 ((( 𝑗𝑥) = (𝑋𝑥) ∧ 𝑗 = 𝐽) → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
119, 10mpancom 422 . . . 4 (𝑗 = 𝐽 → (( 𝑗𝑥) ∈ 𝑗 ↔ (𝑋𝑥) ∈ 𝐽))
128, 11rabeqbidv 2732 . . 3 (𝑗 = 𝐽 → {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
13 df-cld 13228 . . 3 Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 𝑗 ∣ ( 𝑗𝑥) ∈ 𝑗})
1412, 13fvmptg 5587 . 2 ((𝐽 ∈ Top ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽} ∈ V) → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
155, 14mpdan 421 1 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ 𝐽})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2737  cdif 3126  𝒫 cpw 3574   cuni 3807  cfv 5211  Topctop 13128  Clsdccld 13225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-top 13129  df-cld 13228
This theorem is referenced by:  iscld  13236
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