Theorem clsf2 44363

Description: The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 22992. (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 22992	. . 3 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
3		clselmap.k	. . . . 5 ⊢ 𝐾 = (cls‘𝐽)
4	3	feq1i 6653	. . . 4 ⊢ (𝐾:𝒫 𝑋⟶(Clsd‘𝐽) ↔ (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽))
5		df-f 6496	. . . 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 22974	. . . . 5 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
8		sstr2 3940	. . . . 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 6496	. 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 2113   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  Topctop 22837  Clsdccld 22960  clsccl 22962

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-cld 22963  df-cls 22965

This theorem is referenced by:  clselmap  44364