Theorem clsf2 44138

Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 22956. (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 22956	. . 3 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
3		clselmap.k	. . . . 5 ⊢ 𝐾 = (cls‘𝐽)
4	3	feq1i 6638	. . . 4 ⊢ (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
5		df-f 6481	. . . 4 ⊢ (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
6	4, 5	sylbb1 237	. . 3 ⊢ ((cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)))
7	1	cldss2 22938	. . . . 5 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
8		sstr2 3939	. . . . 5 ⊢ (ran 𝐾 ⊆ (Clsd‘𝐽) → ((Clsd‘𝐽) ⊆ 𝒫 𝑋 → ran 𝐾 ⊆ 𝒫 𝑋))
9	7, 8	mpi 20	. . . 4 ⊢ (ran 𝐾 ⊆ (Clsd‘𝐽) → ran 𝐾 ⊆ 𝒫 𝑋)
10	9	anim2i 617	. . 3 ⊢ ((𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽)) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
11	2, 6, 10	3syl 18	. 2 ⊢ (𝐽 ∈ Top → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
12		df-f 6481	. 2 ⊢ (𝐾:𝒫 𝑋⟶𝒫 𝑋 ↔ (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ 𝒫 𝑋))
13	11, 12	sylibr 234	1 ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ⊆ wss 3900  𝒫 cpw 4548  ∪ cuni 4857  ran crn 5615   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477  Topctop 22801  Clsdccld 22924  clsccl 22926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-top 22802  df-cld 22927  df-cls 22929

This theorem is referenced by:  clselmap  44139