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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 44039 | 1 ⊢ (𝜑 → 𝐵 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: ntrclsrcomplex 44048 ntrclsf1o 44064 ntrclsnvobr 44065 ntrclselnel1 44070 ntrclsfv 44072 ntrclscls00 44079 ntrclsiso 44080 ntrclsk2 44081 ntrclskb 44082 ntrclsk3 44083 ntrclsk13 44084 | 
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