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Theorem clsfval 21922
Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1 𝑋 = 𝐽
Assertion
Ref Expression
clsfval (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋
Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem clsfval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4 𝑋 = 𝐽
21topopn 21803 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5271 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 7037 . . 3 (𝒫 𝑋 ∈ V → (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V)
52, 3, 43syl 18 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V)
6 unieq 4830 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2796 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4532 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6717 . . . . . 6 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
109rabeqdv 3395 . . . . 5 (𝑗 = 𝐽 → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
1110inteqd 4864 . . . 4 (𝑗 = 𝐽 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
128, 11mpteq12dv 5140 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
13 df-cls 21918 . . 3 cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
1412, 13fvmptg 6816 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V) → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
155, 14mpdan 687 1 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {crab 3065  Vcvv 3408  wss 3866  𝒫 cpw 4513   cuni 4819   cint 4859  cmpt 5135  cfv 6380  Topctop 21790  Clsdccld 21913  clsccl 21915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-cls 21918
This theorem is referenced by:  clsval  21934  clsf  21945  mrccls  21976
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