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Theorem clsf 21230
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 4390 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval 21219 . . . 4 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) = {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦})
4 fvex 6450 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
53, 4syl6eqelr 2915 . . 3 ((𝐽 ∈ Top ∧ 𝑥𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
61, 5sylan2 586 . 2 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦} ∈ V)
72clsfval 21207 . 2 (𝐽 ∈ Top → (cls‘𝐽) = (𝑥 ∈ 𝒫 𝑋 {𝑦 ∈ (Clsd‘𝐽) ∣ 𝑥𝑦}))
8 elpwi 4390 . . 3 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
92clscld 21229 . . 3 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
108, 9sylan2 586 . 2 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
116, 7, 10fmpt2d 6647 1 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  wss 3798  𝒫 cpw 4380   cuni 4660   cint 4699  wf 6123  cfv 6127  Topctop 21075  Clsdccld 21198  clsccl 21200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-top 21076  df-cld 21201  df-cls 21203
This theorem is referenced by:  clsf2  39263
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