![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneifv1 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the function value of 𝐹 is the neighborhood function. (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 |
---|---|
ntrneifv1 | ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.r | . 2 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
2 | ntrnei.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑𝑚 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
3 | ntrnei.f | . . . . . 6 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
4 | 2, 3, 1 | ntrneif1o 39910 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵)) |
5 | f1ofn 6484 | . . . . 5 ⊢ (𝐹:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑𝑚 𝐵) → 𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
7 | 2, 3, 1 | ntrneiiex 39911 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
8 | 6, 7 | jca 512 | . . 3 ⊢ (𝜑 → (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))) |
9 | fnfun 6323 | . . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → Fun 𝐹) | |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → Fun 𝐹) |
11 | fndm 6325 | . . . . . 6 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | |
12 | 11 | eleq2d 2868 | . . . . 5 ⊢ (𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐹 ↔ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))) |
13 | 12 | biimpar 478 | . . . 4 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐹) |
14 | 10, 13 | jca 512 | . . 3 ⊢ ((𝐹 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹)) |
15 | funbrfvb 6588 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹) → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁)) | |
16 | 8, 14, 15 | 3syl 18 | . 2 ⊢ (𝜑 → ((𝐹‘𝐼) = 𝑁 ↔ 𝐼𝐹𝑁)) |
17 | 1, 16 | mpbird 258 | 1 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {crab 3109 Vcvv 3437 𝒫 cpw 4453 class class class wbr 4962 ↦ cmpt 5041 dom cdm 5443 Fun wfun 6219 Fn wfn 6220 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 ↑𝑚 cmap 8256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-map 8258 |
This theorem is referenced by: ntrneiel 39916 |
Copyright terms: Public domain | W3C validator |