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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 5696 | . 2 ⊢ ran 𝐼 = ran (int‘𝐽) |
3 | vpwex 5176 | . . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V | |
4 | 3 | inex2 5120 | . . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
5 | 4 | uniex 7330 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
6 | 5 | rgenw 3119 | . . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
7 | nfcv 2951 | . . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋 | |
8 | 7 | fnmptf 6359 | . . . . 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 21320 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
12 | 11 | fneq1d 6323 | . . . 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 4469 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋) | |
15 | 10 | ntropn 21345 | . . . . . 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 3150 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) |
19 | fnfvrnss 6754 | . . 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 3942 | 1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ∀wral 3107 Vcvv 3440 ∩ cin 3864 ⊆ wss 3865 𝒫 cpw 4459 ∪ cuni 4751 ↦ cmpt 5047 ran crn 5451 Fn wfn 6227 ‘cfv 6232 Topctop 21189 intcnt 21313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-top 21190 df-ntr 21316 |
This theorem is referenced by: ntrf 39979 |
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