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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrf | Structured version Visualization version GIF version | ||
| Description: The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.) |
| Ref | Expression |
|---|---|
| ntrrn.x | ⊢ 𝑋 = ∪ 𝐽 |
| ntrrn.i | ⊢ 𝐼 = (int‘𝐽) |
| Ref | Expression |
|---|---|
| ntrf | ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vpwex 5377 | . . . . . 6 ⊢ 𝒫 𝑠 ∈ V | |
| 2 | 1 | inex2 5318 | . . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
| 3 | 2 | uniex 7761 | . . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
| 4 | eqid 2737 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) | |
| 5 | 3, 4 | fnmpti 6711 | . . 3 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋 |
| 6 | ntrrn.i | . . . . 5 ⊢ 𝐼 = (int‘𝐽) | |
| 7 | ntrrn.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
| 8 | 7 | ntrfval 23032 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
| 9 | 6, 8 | eqtrid 2789 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
| 10 | 9 | fneq1d 6661 | . . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
| 11 | 5, 10 | mpbiri 258 | . 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋) |
| 12 | 7, 6 | ntrrn 44135 | . 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
| 13 | df-f 6565 | . 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽)) | |
| 14 | 11, 12, 13 | sylanbrc 583 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ↦ cmpt 5225 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Topctop 22899 intcnt 23025 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-ntr 23028 |
| This theorem is referenced by: ntrf2 44137 |
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