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Theorem 0res 30373
 Description: Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Assertion
Ref Expression
0res (∅ ↾ 𝐴) = ∅

Proof of Theorem 0res
StepHypRef Expression
1 df-res 5555 . 2 (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2 0in 4330 . 2 (∅ ∩ (𝐴 × V)) = ∅
31, 2eqtri 2847 1 (∅ ↾ 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3480   ∩ cin 3918  ∅c0 4276   × cxp 5541   ↾ cres 5545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-nul 4277  df-res 5555 This theorem is referenced by:  cycpmrn  30827  tocyccntz  30828
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