Theorem 0res 32680

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 5636	. 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2		0in 4349	. 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅
3	1, 2	eqtri 2759	1 ⊢ (∅ ↾ 𝐴) = ∅

Syntax hints:   = wceq 1541  Vcvv 3440   ∩ cin 3900  ∅c0 4285   × cxp 5622   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-in 3908  df-nul 4286  df-res 5636

This theorem is referenced by:  cycpmrn  33227  tocyccntz  33228