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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 41377 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
5 | 2, 3, 1 | ntrneinex 41379 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
6 | dff1o3 6676 | . . . . . . . 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 500 | . . . . . . 7 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Fun ◡𝐹) |
8 | 7 | adantr 484 | . . . . . 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 5571 | . . . . . . . . 9 ⊢ ran 𝐹 = dom ◡𝐹 | |
10 | f1ofo 6677 | . . . . . . . . . 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 6645 | . . . . . . . . . 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 2793 | . . . . . . . 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 2824 | . . . . . . 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 481 | . . . . . 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 515 | . . . . 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 587 | . . . 4 ⊢ (𝜑 → (Fun ◡𝐹 ∧ 𝑁 ∈ dom ◡𝐹)) |
18 | funbrfvb 6776 | . . . 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 41378 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
21 | brcnvg 5757 | . . . 4 ⊢ ((𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁)) | |
22 | 5, 20, 21 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁◡𝐹𝐼 ↔ 𝐼𝐹𝑁)) |
23 | 19, 22 | bitrd 282 | . 2 ⊢ (𝜑 → ((◡𝐹‘𝑁) = 𝐼 ↔ 𝐼𝐹𝑁)) |
24 | 1, 23 | mpbird 260 | 1 ⊢ (𝜑 → (◡𝐹‘𝑁) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {crab 3066 Vcvv 3415 𝒫 cpw 4522 class class class wbr 5062 ↦ cmpt 5144 ◡ccnv 5559 dom cdm 5560 ran crn 5561 Fun wfun 6383 –onto→wfo 6387 –1-1-onto→wf1o 6388 ‘cfv 6389 (class class class)co 7222 ∈ cmpo 7224 ↑m cmap 8517 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-1st 7770 df-2nd 7771 df-map 8519 |
This theorem is referenced by: (None) |
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