Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneicnv | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then converse of 𝐹 is known. (Contributed by RP, 29-May-2021.) |
Ref | Expression |
---|---|
ntrnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
ntrnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
ntrnei.r | ⊢ (𝜑 → 𝐼𝐹𝑁) |
Ref | Expression |
---|---|
ntrneicnv | ⊢ (𝜑 → ◡𝐹 = (𝐵𝑂𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.o | . 2 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | ntrnei.f | . . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
3 | ntrnei.r | . . . 4 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
4 | 1, 2, 3 | ntrneibex 41312 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 4 | pwexd 5261 | . 2 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
6 | eqid 2734 | . 2 ⊢ (𝐵𝑂𝒫 𝐵) = (𝐵𝑂𝒫 𝐵) | |
7 | 1, 5, 4, 2, 6 | fsovcnvd 41251 | 1 ⊢ (𝜑 → ◡𝐹 = (𝐵𝑂𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 𝒫 cpw 4503 class class class wbr 5043 ↦ cmpt 5124 ◡ccnv 5539 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 ↑m cmap 8497 |
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 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
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 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |