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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 5553 | . 2 ⊢ ran 𝐼 = ran (int‘𝐽) |
3 | vpwex 5045 | . . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V | |
4 | 3 | inex2 4993 | . . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
5 | 4 | uniex 7185 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
6 | 5 | rgenw 3103 | . . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
7 | nfcv 2939 | . . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋 | |
8 | 7 | fnmptf 6225 | . . . . 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 21154 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
12 | 11 | fneq1d 6190 | . . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
13 | 9, 12 | mpbird 249 | . . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋) |
14 | elpwi 4357 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋) | |
15 | 10 | ntropn 21179 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽) |
16 | 15 | ex 402 | . . . . 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 3144 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) |
19 | fnfvrnss 6614 | . . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽) | |
20 | 13, 18, 19 | syl2anc 580 | . 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽) |
21 | 2, 20 | syl5eqss 3843 | 1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∀wral 3087 Vcvv 3383 ∩ cin 3766 ⊆ wss 3767 𝒫 cpw 4347 ∪ cuni 4626 ↦ cmpt 4920 ran crn 5311 Fn wfn 6094 ‘cfv 6099 Topctop 21023 intcnt 21147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-top 21024 df-ntr 21150 |
This theorem is referenced by: ntrf 39191 |
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