Theorem 0res 30367

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 5531	. 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2		0in 4301	. 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅
3	1, 2	eqtri 2821	1 ⊢ (∅ ↾ 𝐴) = ∅

Syntax hints:   = wceq 1538  Vcvv 3441   ∩ cin 3880  ∅c0 4243   × cxp 5517   ↾ cres 5521

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-nul 4244  df-res 5531

This theorem is referenced by:  cycpmrn  30835  tocyccntz  30836