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Theorem clsf 21667
 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 4506 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval 21656 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
4 fvex 6663 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
53, 4eqeltrrdi 2899 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
61, 5sylan2 595 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
72clsfval 21644 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
8 elpwi 4506 . . 3 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
92clscld 21666 . . 3 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
108, 9sylan2 595 . 2 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
116, 7, 10fmpt2d 6869 1 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4801  ∩ cint 4839  ⟶wf 6323  ‘cfv 6327  Topctop 21512  Clsdccld 21635  clsccl 21637 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-top 21513  df-cld 21638  df-cls 21640 This theorem is referenced by:  clsf2  40893
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