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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 5810 | . 2 ⊢ ran 𝐼 = ran (int‘𝐽) |
3 | vpwex 5281 | . . . . . . . 8 ⊢ 𝒫 𝑠 ∈ V | |
4 | 3 | inex2 5225 | . . . . . . 7 ⊢ (𝐽 ∩ 𝒫 𝑠) ∈ V |
5 | 4 | uniex 7470 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
6 | 5 | rgenw 3153 | . . . . 5 ⊢ ∀𝑠 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑠) ∈ V |
7 | nfcv 2980 | . . . . . 6 ⊢ Ⅎ𝑠𝒫 𝑋 | |
8 | 7 | fnmptf 6487 | . . . . 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 21635 | . . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠))) |
12 | 11 | fneq1d 6449 | . . . 4 ⊢ (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)) |
13 | 9, 12 | mpbird 259 | . . 3 ⊢ (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋) |
14 | elpwi 4551 | . . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋) | |
15 | 10 | ntropn 21660 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ 𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽) |
16 | 15 | ex 415 | . . . . 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 3184 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) |
19 | fnfvrnss 6887 | . . 3 ⊢ (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽) | |
20 | 13, 18, 19 | syl2anc 586 | . 2 ⊢ (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽) |
21 | 2, 20 | eqsstrid 4018 | 1 ⊢ (𝐽 ∈ Top → ran 𝐼 ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4542 ∪ cuni 4841 ↦ cmpt 5149 ran crn 5559 Fn wfn 6353 ‘cfv 6358 Topctop 21504 intcnt 21628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-top 21505 df-ntr 21631 |
This theorem is referenced by: ntrf 40479 |
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