Theorem res0 3377

Description: A restriction to the empty set is empty.

Assertion

Ref	Expression
res0	⊢ (A ↾ ∅) = ∅

Proof of Theorem res0

Step	Hyp	Ref	Expression
1		df-res 3196	. 2 ⊢ (A ↾ ∅) = (A ∩ (∅ × V))
2		xp0r 3245	. . 3 ⊢ (∅ × V) = ∅
3	2	ineq2i 2217	. 2 ⊢ (A ∩ (∅ × V)) = (A ∩ ∅)
4		in0 2302	. 2 ⊢ (A ∩ ∅) = ∅
5	1, 3, 4	3eqtr 1502	1 ⊢ (A ↾ ∅) = ∅

Syntax hints:   = wceq 958  Vcvv 1814   ∩ cin 2049  ∅c0 2283   × cxp 3174   ↾ cres 3178

This theorem is referenced by:  ima0 3426  tz7.44-1 3934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-res 3196

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