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Theorem clsfval 22949
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 22828 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5382 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 7239 . . 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 4923 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1eqtr4di 2786 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4623 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6902 . . . . . 6 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
109rabeqdv 3446 . . . . 5 (𝑗 = 𝐽 → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
1110inteqd 4958 . . . 4 (𝑗 = 𝐽 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
128, 11mpteq12dv 5243 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
13 df-cls 22945 . . 3 cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
1412, 13fvmptg 7008 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V) → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
155, 14mpdan 685 1 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3430  Vcvv 3473  wss 3949  𝒫 cpw 4606   cuni 4912   cint 4953  cmpt 5235  cfv 6553  Topctop 22815  Clsdccld 22940  clsccl 22942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-top 22816  df-cls 22945
This theorem is referenced by:  clsval  22961  clsf  22972  mrccls  23003
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