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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrrn | Structured version Visualization version GIF version |
Description: The range of the interior function of a topology a subset of 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 |
---|---|
ntrrn | ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrrn.i | . . 3 ⊢ 𝐼 = (int‘𝐽) | |
2 | 1 | rneqi 5800 | . 2 ⊢ ran 𝐼 = ran (int‘𝐽) |
3 | vpwex 5269 | . . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V | |
4 | 3 | inex2 5213 | . . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
5 | 4 | uniex 7454 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
6 | 5 | rgenw 3147 | . . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
7 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋 | |
8 | 7 | fnmptf 6477 | . . . . 5 ⊢ (∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋) |
9 | 6, 8 | mp1i 13 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋) |
10 | ntrrn.x | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
11 | 10 | ntrfval 21560 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
12 | 11 | fneq1d 6439 | . . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
13 | 9, 12 | mpbird 258 | . . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋) |
14 | elpwi 4547 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋) | |
15 | 10 | ntropn 21585 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽) |
16 | 15 | ex 413 | . . . . 5 ⊢ (𝐽 ∈ Top → (𝑠 ⊆ 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽)) |
17 | 14, 16 | syl5 34 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽)) |
18 | 17 | ralrimiv 3178 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) |
19 | fnfvrnss 6876 | . . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽) | |
20 | 13, 18, 19 | syl2anc 584 | . 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽) |
21 | 2, 20 | eqsstrid 4012 | 1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ↦ cmpt 5137 ran crn 5549 Fn wfn 6343 ‘cfv 6348 Topctop 21429 intcnt 21553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-top 21430 df-ntr 21556 |
This theorem is referenced by: ntrf 40351 |
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