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Theorem clsf 20762
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 4140 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval 20751 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
4 fvex 6158 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
53, 4syl6eqelr 2707 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
61, 5sylan2 491 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
72clsfval 20739 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
8 elpwi 4140 . . 3 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
92clscld 20761 . . 3 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
108, 9sylan2 491 . 2 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
116, 7, 10fmpt2d 6348 1 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   cuni 4402   cint 4440  wf 5843  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-cld 20733  df-cls 20735
This theorem is referenced by:  clsf2  37906
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