Theorem 0res 33197

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 5662	. 2 ⊢ (∅ ↾ 𝐴) = (∅ ∩ (𝐴 × V))
2		0in 4346	. 2 ⊢ (∅ ∩ (𝐴 × V)) = ∅
3	1, 2	eqtri 2783	1 ⊢ (∅ ↾ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450   ∩ cin 4089  ∅c0 4278   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 4047  df-in 4091  df-nul 4279  df-res 5662

This theorem is used by:  cycpmrn  33704  tocyccntz  33705