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Theorem clsf 21584
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 4547 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval 21573 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
4 fvex 6676 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
53, 4syl6eqelr 2919 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
61, 5sylan2 592 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
72clsfval 21561 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
8 elpwi 4547 . . 3 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
92clscld 21583 . . 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 6879 1 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535   cuni 4830   cint 4867  wf 6344  cfv 6348  Topctop 21429  Clsdccld 21552  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-cls 21557
This theorem is referenced by:  clsf2  40354
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