![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneifv2 | 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 converse of 𝐹 is the interior 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 |
---|---|
ntrneifv2 | ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼) |
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 44065 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
5 | 2, 3, 1 | ntrneinex 44067 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
6 | dff1o3 6855 | . . . . . . . 8 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ↔ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ Fun ◡𝐹)) | |
7 | 6 | simprbi 496 | . . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Fun ◡𝐹) |
8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → Fun ◡𝐹) |
9 | df-rn 5700 | . . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹 | |
10 | f1ofo 6856 | . . . . . . . . . 10 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) | |
11 | forn 6824 | . . . . . . . . . 10 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → ran 𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)) |
13 | 9, 12 | eqtr3id 2789 | . . . . . . . 8 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → dom ◡𝐹 = (𝒫 𝒫 𝐵 ↑m 𝐵)) |
14 | 13 | eleq2d 2825 | . . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → (𝑁 ∈ dom ◡𝐹 ↔ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))) |
15 | 14 | biimpar 477 | . . . . . 6 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → 𝑁 ∈ dom ◡𝐹) |
16 | 8, 15 | jca 511 | . . . . 5 ⊢ ((𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹)) |
17 | 4, 5, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹)) |
18 | funbrfvb 6962 | . . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹) → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼)) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝑁◡𝐹𝐼)) |
20 | 2, 3, 1 | ntrneiiex 44066 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
21 | brcnvg 5893 | . . . 4 ⊢ ((𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁)) | |
22 | 5, 20, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁)) |
23 | 19, 22 | bitrd 279 | . 2 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝐼𝐹𝑁)) |
24 | 1, 23 | mpbird 257 | 1 ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 ran crn 5690 Fun wfun 6557 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |