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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 5359 | . . . . . 6 ⊢ 𝒫 𝑠 ∈ V | |
2 | 1 | inex2 5302 | . . . . 5 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
3 | 2 | uniex 7705 | . . . 4 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
4 | eqid 2731 | . . . 4 ⊢ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) | |
5 | 3, 4 | fnmpti 6671 | . . 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 22434 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
9 | 6, 8 | eqtrid 2783 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
10 | 9 | fneq1d 6622 | . . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
11 | 5, 10 | mpbiri 257 | . 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋) |
12 | 7, 6 | ntrrn 42556 | . 2 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
13 | df-f 6527 | . 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 1541 ∈ wcel 2106 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4587 ∪ cuni 4892 ↦ cmpt 5215 ran crn 5661 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 Topctop 22301 intcnt 22427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-top 22302 df-ntr 22430 |
This theorem is referenced by: ntrf2 42558 |
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