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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 5806 | . 2 ⊢ ran 𝐼 = ran (int‘𝐽) |
3 | vpwex 5270 | . . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V | |
4 | 3 | inex2 5211 | . . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
5 | 4 | uniex 7529 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
6 | 5 | rgenw 3073 | . . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
7 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋 | |
8 | 7 | fnmptf 6514 | . . . . 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 21921 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
12 | 11 | fneq1d 6472 | . . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
13 | 9, 12 | mpbird 260 | . . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋) |
14 | elpwi 4522 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋) | |
15 | 10 | ntropn 21946 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽) |
16 | 15 | ex 416 | . . . . 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 3104 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) |
19 | fnfvrnss 6937 | . . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽) | |
20 | 13, 18, 19 | syl2anc 587 | . 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽) |
21 | 2, 20 | eqsstrid 3949 | 1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 𝒫 cpw 4513 ∪ cuni 4819 ↦ cmpt 5135 ran crn 5552 Fn wfn 6375 ‘cfv 6380 Topctop 21790 intcnt 21914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-top 21791 df-ntr 21917 |
This theorem is referenced by: ntrf 41410 |
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