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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 40369 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ‘cfv 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 ax-pow 5258 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 |
This theorem is referenced by: ntrclsrcomplex 40378 ntrclsf1o 40394 ntrclsnvobr 40395 ntrclselnel1 40400 ntrclsfv 40402 ntrclscls00 40409 ntrclsiso 40410 ntrclsk2 40411 ntrclskb 40412 ntrclsk3 40413 ntrclsk13 40414 |
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