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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 5255 | . . . . . 6 ⊢ 𝒫 𝑠 ∈ V | |
2 | 1 | inex2 5196 | . . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
3 | 2 | uniex 7507 | . . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
4 | eqid 2736 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) | |
5 | 3, 4 | fnmpti 6499 | . . 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 21875 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
9 | 6, 8 | syl5eq 2783 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
10 | 9 | fneq1d 6450 | . . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
11 | 5, 10 | mpbiri 261 | . 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋) |
12 | 7, 6 | ntrrn 41350 | . 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
13 | df-f 6362 | . 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽)) | |
14 | 11, 12, 13 | sylanbrc 586 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 ↦ cmpt 5120 ran crn 5537 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 Topctop 21744 intcnt 21868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-top 21745 df-ntr 21871 |
This theorem is referenced by: ntrf2 41352 |
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