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

Theorem clsfval 21050
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 20931 . . 3 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4980 . . 3 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 mptexg 6628 . . 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 4582 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝐽)
76, 1syl6eqr 2823 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝑋)
87pweqd 4302 . . . 4 (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝑋)
9 fveq2 6332 . . . . . 6 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
10 rabeq 3342 . . . . . 6 ((Clsd‘𝑗) = (Clsd‘𝐽) → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
119, 10syl 17 . . . . 5 (𝑗 = 𝐽 → {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
1211inteqd 4616 . . . 4 (𝑗 = 𝐽 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦} = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
138, 12mpteq12dv 4867 . . 3 (𝑗 = 𝐽 → (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
14 df-cls 21046 . . 3 cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
1513, 14fvmptg 6422 . 2 ((𝐽 ∈ Top ∧ (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}) ∈ V) → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
165, 15mpdan 667 1 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4297   cuni 4574   cint 4611  cmpt 4863  cfv 6031  Topctop 20918  Clsdccld 21041  clsccl 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20919  df-cls 21046
This theorem is referenced by:  clsval  21062  clsf  21073  mrccls  21104
  Copyright terms: Public domain W3C validator