Theorem clsf2 44872

Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 23205. (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

Step	Hyp	Ref	Expression
1		clselmap.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	clsf 23205	. . 3 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
3		clselmap.k	. . . . 5 ⊢ 𝐾 = (cls‘𝐽)
4	3	feq1i 6696	. . . 4 ⊢ (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
5		df-f 6540	. . . 4 ⊢ (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
6	4, 5	sylbb1 240	. . 3 ⊢ ((cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
7	1	cldss2 23187	. . . . 5 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
8		sstr2 3944	. . . . 5 ⊢ (ran 𝐾 ⊆ (Clsd‘𝐽) → ((Clsd‘𝐽) ⊆ 𝒫 𝑋 → ran 𝐾 ⊆ 𝒫 𝑋))
9	7, 8	mpi 21	. . . 4 ⊢ (ran 𝐾 ⊆ (Clsd‘𝐽) → ran 𝐾 ⊆ 𝒫 𝑋)
10	9	anim2i 628	. . 3 ⊢ ((𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
11	2, 6, 10	3syl 19	. 2 ⊢ (𝐽 ∈ Top → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
12		df-f 6540	. 2 ⊢ (𝐾:𝒫 𝑋⟶𝒫 𝑋 ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
13	11, 12	sylibr 237	1 ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872  ran crn 5662   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  Topctop 23050  Clsdccld 23173  clsccl 23175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 23051  df-cld 23176  df-cls 23178

This theorem is referenced by:  clselmap  44873