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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 5383 | . . . . . 6 ⊢ 𝒫 𝑠 ∈ V | |
2 | 1 | inex2 5324 | . . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
3 | 2 | uniex 7760 | . . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
4 | eqid 2735 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) | |
5 | 3, 4 | fnmpti 6712 | . . 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 23048 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
9 | 6, 8 | eqtrid 2787 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
10 | 9 | fneq1d 6662 | . . 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 44112 | . 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
13 | df-f 6567 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Topctop 22915 intcnt 23041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-top 22916 df-ntr 23044 |
This theorem is referenced by: ntrf2 44114 |
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