| Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneiiex | Structured version Visualization version GIF version | ||
| Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior function exists. (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 |
|---|---|
| ntrneiiex | ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ntrnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
| 2 | ntrnei.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
| 3 | ntrnei.r | . . . . 5 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
| 4 | 1, 2, 3 | ntrneif1o 44088 | . . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
| 5 | f1orel 6851 | . . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Rel 𝐹) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
| 7 | releldm 5955 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹) | |
| 8 | 6, 3, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹) |
| 9 | f1odm 6852 | . . 3 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
| 10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| 11 | 8, 10 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 Rel wrel 5690 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: ntrneifv1 44092 ntrneifv2 44093 ntrneiel 44094 ntrneifv4 44098 ntrneiel2 44099 ntrneicls00 44102 ntrneicls11 44103 ntrneiiso 44104 ntrneik2 44105 ntrneikb 44107 ntrneixb 44108 ntrneik3 44109 ntrneix3 44110 ntrneik13 44111 ntrneix13 44112 ntrneik4w 44113 ntrneik4 44114 |
| Copyright terms: Public domain | W3C validator |