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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsbex | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.) |
Ref | Expression |
---|---|
ntrclsbex.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrclsbex.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclsbex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrclsbex.r | . 2 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
2 | ntrclsbex.d | . . 3 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝐷 = (𝑂‘𝐵)) |
4 | 1, 3 | brfvimex 44016 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 |
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-nul 5312 ax-pr 5438 |
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-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: ntrclsrcomplex 44025 ntrclsf1o 44041 ntrclsnvobr 44042 ntrclselnel1 44047 ntrclsfv 44049 ntrclscls00 44056 ntrclsiso 44057 ntrclsk2 44058 ntrclskb 44059 ntrclsk3 44060 ntrclsk13 44061 |
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