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Theorem clsf2 44029
Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 23070. (Contributed by RP, 22-Apr-2021.)
Hypotheses
Ref Expression
clselmap.x 𝑋 = 𝐽
clselmap.k 𝐾 = (cls‘𝐽)
Assertion
Ref Expression
clsf2 (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)

Proof of Theorem clsf2
StepHypRef Expression
1 clselmap.x . . . 4 𝑋 = 𝐽
21clsf 23070 . . 3 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
3 clselmap.k . . . . 5 𝐾 = (cls‘𝐽)
43feq1i 6737 . . . 4 (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
5 df-f 6576 . . . 4 (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
64, 5sylbb1 237 . . 3 ((cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
71cldss2 23052 . . . . 5 (Clsd‘𝐽) ⊆ 𝒫 𝑋
8 sstr2 4009 . . . . 5 (ran 𝐾 ⊆ (Clsd‘𝐽) → ((Clsd‘𝐽) ⊆ 𝒫 𝑋 → ran 𝐾 ⊆ 𝒫 𝑋))
97, 8mpi 20 . . . 4 (ran 𝐾 ⊆ (Clsd‘𝐽) → ran 𝐾 ⊆ 𝒫 𝑋)
109anim2i 616 . . 3 ((𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
112, 6, 103syl 18 . 2 (𝐽 ∈ Top → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
12 df-f 6576 . 2 (𝐾:𝒫 𝑋⟶𝒫 𝑋 ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
1311, 12sylibr 234 1 (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  wss 3970  𝒫 cpw 4622   cuni 4931  ran crn 5700   Fn wfn 6567  wf 6568  cfv 6572  Topctop 22913  Clsdccld 23038  clsccl 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-top 22914  df-cld 23041  df-cls 23043
This theorem is referenced by:  clselmap  44030
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