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Theorem clsf 22876
Description: The closure function is a function from subsets of the base to closed sets. (Contributed by Mario Carneiro, 11-Apr-2015.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsf (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))

Proof of Theorem clsf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4602 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval 22865 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
4 fvex 6895 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
53, 4eqeltrrdi 2834 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
61, 5sylan2 592 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
72clsfval 22853 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
8 elpwi 4602 . . 3 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
92clscld 22875 . . 3 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
108, 9sylan2 592 . 2 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
116, 7, 10fmpt2d 7116 1 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {crab 3424  Vcvv 3466  wss 3941  𝒫 cpw 4595   cuni 4900   cint 4941  wf 6530  cfv 6534  Topctop 22719  Clsdccld 22844  clsccl 22846
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22720  df-cld 22847  df-cls 22849
This theorem is referenced by:  clsf2  43391
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