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

Theorem clsfval 21317
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 21198 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 5170 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6850 . . 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 4753 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2849 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4458 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6538 . . . . . 6 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
109rabeqdv 3429 . . . . 5 (𝑗 = 𝐽 → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
1110inteqd 4787 . . . 4 (𝑗 = 𝐽 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
128, 11mpteq12dv 5045 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
13 df-cls 21313 . . 3 cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
1412, 13fvmptg 6633 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V) → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
155, 14mpdan 683 1 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  {crab 3109  Vcvv 3437  wss 3859  𝒫 cpw 4453   cuni 4745   cint 4782  cmpt 5041  cfv 6225  Topctop 21185  Clsdccld 21308  clsccl 21310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-cls 21313
This theorem is referenced by:  clsval  21329  clsf  21340  mrccls  21371
  Copyright terms: Public domain W3C validator