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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 4775 | . . . . . 6 ⊢ 𝒫 𝑠 ∈ V | |
2 | 1 | inex2 4728 | . . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
3 | 2 | uniex 6851 | . . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
4 | eqid 2610 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) | |
5 | 3, 4 | fnmpti 5935 | . . 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 20638 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
9 | 6, 8 | syl5eq 2656 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
10 | 9 | fneq1d 5895 | . . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
11 | 5, 10 | mpbiri 247 | . 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋) |
12 | 7, 6 | ntrrn 37440 | . 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
13 | df-f 5808 | . 2 ⊢ (𝐼:𝒫 𝑋⟶𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 ⊆ 𝐽)) | |
14 | 11, 12, 13 | sylanbrc 695 | 1 ⊢ (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ↦ cmpt 4643 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Topctop 20517 intcnt 20631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-top 20521 df-ntr 20634 |
This theorem is referenced by: ntrf2 37442 |
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