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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsf2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
clselmap.x | ⊢ 𝑋 = ∪ 𝐽 |
clselmap.k | ⊢ 𝐾 = (cls‘𝐽) |
Ref | Expression |
---|---|
clsf2 | ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clselmap.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | clsf 23070 | . . 3 ⊢ (𝐽 ∈ Top → (cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽)) |
3 | clselmap.k | . . . . 5 ⊢ 𝐾 = (cls‘𝐽) | |
4 | 3 | feq1i 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))) | |
6 | 4, 5 | sylbb1 237 | . . 3 ⊢ ((cls‘𝐽):𝒫 𝑋⟶(Clsd‘𝐽) → (𝐾 Fn 𝒫 𝑋 ∧ ran 𝐾 ⊆ (Clsd‘𝐽))) |
7 | 1 | cldss2 23052 | . . . . 5 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋 |
8 | sstr2 4009 | . . . . 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 616 | . . 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 6576 | . 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) |
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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