Theorem 0res 32538

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 5652	. 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2		0in 4362	. 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅
3	1, 2	eqtri 2753	1 ⊢ (∅ ↾ 𝐴) = ∅

Syntax hints:   = wceq 1540  Vcvv 3450   ∩ cin 3915  ∅c0 4298   × cxp 5638   ↾ cres 5642

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-in 3923  df-nul 4299  df-res 5652

This theorem is referenced by:  cycpmrn  33106  tocyccntz  33107