Theorem 0res 33266

Description: Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)

Assertion

Ref	Expression
0res	⊢ (∅ ↾ 𝐴) = ∅

Proof of Theorem 0res

Step	Hyp	Ref	Expression
1		df-res 5667	. 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2		0in 4350	. 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅
3	1, 2	eqtri 2786	1 ⊢ (∅ ↾ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∩ cin 4092  ∅c0 4281   × cxp 5653   ↾ cres 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094  df-nul 4282  df-res 5667

This theorem is used by:  cycpmrn  33771  tocyccntz  33772