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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 40500 | . . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
5 | f1orel 6611 | . . . 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 5807 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹) | |
8 | 6, 3, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹) |
9 | f1odm 6612 | . . 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 2914 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3141 Vcvv 3491 𝒫 cpw 4532 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5548 Rel wrel 5553 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 ↑m cmap 8399 |
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 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-map 8401 |
This theorem is referenced by: ntrneifv1 40504 ntrneifv2 40505 ntrneiel 40506 ntrneifv4 40510 ntrneiel2 40511 ntrneicls00 40514 ntrneicls11 40515 ntrneiiso 40516 ntrneik2 40517 ntrneikb 40519 ntrneixb 40520 ntrneik3 40521 ntrneix3 40522 ntrneik13 40523 ntrneix13 40524 ntrneik4w 40525 ntrneik4 40526 |
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