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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 22992. (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 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) |
| Colors of variables: wff setvar class |
| 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 |
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