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Theorem clsf2 37941
Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 20775. (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 20775 . . 3 (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
3 clselmap.k . . . . 5 𝐾 = (cls‘𝐽)
43feq1i 5998 . . . 4 (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
5 df-f 5856 . . . 4 (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
64, 5sylbb1 227 . . 3 ((cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
71cldss2 20757 . . . . 5 (Clsd‘𝐽) ⊆ 𝒫 𝑋
8 sstr2 3594 . . . . 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 592 . . 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 5856 . 2 (𝐾:𝒫 𝑋⟶𝒫 𝑋 ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
1311, 12sylibr 224 1 (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3559  𝒫 cpw 4135   cuni 4407  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  Topctop 20630  Clsdccld 20743  clsccl 20745
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-top 20631  df-cld 20746  df-cls 20748
This theorem is referenced by:  clselmap  37942
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