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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0res | Structured version Visualization version GIF version |
Description: Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
0res | ⊢ (∅ ↾ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5686 | . 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V)) | |
2 | 0in 4391 | . 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅ | |
3 | 1, 2 | eqtri 2754 | 1 ⊢ (∅ ↾ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 Vcvv 3462 ∩ cin 3945 ∅c0 4322 × cxp 5672 ↾ cres 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3949 df-in 3953 df-nul 4323 df-res 5686 |
This theorem is referenced by: cycpmrn 33025 tocyccntz 33026 |
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